Analytical Methods for Heat Transfer and Fluid Flow Problems [1 ed.] 3540222472, 9783540222477

Table of contents :
10.1007/978-3-540-68466-4......Page 1
Analytical Methods for Heat Transfer and Fluid Flow Problems......Page 4
Copyright Page - ISBN: 3540222472......Page 5
Preface......Page 8
Table of Contents......Page 12
List of Symbols......Page 15
List of Copyrighted Figures......Page 20
1 Introduction......Page 23
Problems......Page 31
2.1 Classification of Second-Order Partial Differential Equations......Page 33
2.2.1 Parabolic Second-Order Equations......Page 41
2.2.2 Elliptic Second-Order Equations......Page 43
2.2.3 Hyperbolic Second-Order Equations......Page 45
2.3.1 One-Dimensional Transient Heat Conduction in a Flat Plate......Page 46
2.3.2 Steady-State Heat Conduction in a Rectangular Plate......Page 53
2.3.3 Separation of Variables for the General Case of a Linear Second-Order Partial Differential Equation......Page 59
Problems......Page 61
3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature......Page 65
3.1.1 Velocity Distribution of Hydrodynamically Fully Developed Pipe and Channel Flows......Page 66
3.1.2 Thermal Entrance Solutions for Constant Wall Temperature......Page 68
Properties of the Sturm-Liouville System......Page 72
1. The Problem is Self-Adjoint and Positive Definite......Page 73
2. Eigenvalues and Eigenfunctions......Page 75
4. Eigenfunction Expansions for an Arbitrary Function......Page 76
Laminar Flows......Page 80
Turbulent Flows......Page 82
Heat Transfer in Turbulent Pipe Flow......Page 85
3.2 Thermal Entrance Solutions for an Arbitrary Wall Temperature Distribution......Page 88
3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux......Page 91
3.3.1 Velocity Distribution for the Hydrodynamically Fully Developed Flow in an Axial Rotating Pipe......Page 94
3.3.2 Thermal Entrance Solution for Constant Wall Heat Flux......Page 97
Temperature Distribution for the Fully Developed Flow ( Θ1 )......Page 99
The Temperature Distribution Θ2......Page 100
Problems......Page 105
4 Analytical Solutions for Sturm - Liouville Systems with Large Eigenvalues......Page 109
4.1 Heat Transfer in Turbulent Pipe Flow with Constant Wall Temperature......Page 125
4.2 Heat Transfer in an Axially Rotating Pipe with Constant Wall Temperature......Page 131
4.3 Asymptotic Expressions for other Thermal Boundary Conditions......Page 135
Problems......Page 137
5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)......Page 139
5.1 Heat Transfer for Constant Wall Temperatures for x≤ 0 and x > 0......Page 142
Laminar Pipe Flow......Page 153
Laminar Channel Flow......Page 157
5.1.2 Heat Transfer in Turbulent Pipe and Channel Flows for Small Peclet Numbers......Page 160
Turbulent Pipe Flow......Page 161
5.2 Heat Transfer for Constant Wall Heat Flux for x ≤ 0 and x > 0......Page 164
5.2.1 Heat Transfer in Laminar Pipe and Channel Flows for Small Peclet Numbers......Page 169
5.2.2 Heat Transfer in Turbulent Pipe and Channel Flows for Small Peclet Numbers......Page 172
5.3 Results for Heating Sections with a Finite Length......Page 175
5.3.1 Piecewise Constant Wall Temperature......Page 176
5.3.1 Piecewise Constant Wall Heat Flux......Page 179
5.4 Application of the Solution Method to Related Problems......Page 181
Problems......Page 183
6.1 The Method of Separation of Variables......Page 187
6.2 Transformations Resulting in Linear Partial Differential Equations......Page 194
6.3.1 Incompressible Flow over a Heated Flat Plate......Page 196
6.3.2 Compressible Flow over a Flat, Heated Plate......Page 199
6.4.1 Similarity Solutions for a Transient Heat Conduction Problem......Page 201
6.4.2 Similarity Solutions of the Boundary Layer Equations for Laminar Free Convection Flow on a Vertical Flat Plate......Page 208
6.4.3 Similarity Solutions of the Compressible Boundary Layer Equations......Page 214
Problems......Page 220
Appendix A: The Fully Developed Velocity Profile for Turbulent Duct Flows......Page 225
Appendix B: The Fully Developed Velocity Profile in an Axially Rotating Pipe......Page 237
Appendix C: A Numerical Solution Method for Eigenvalue Problems......Page 249
C.1 Numerical Tools......Page 251
D.1 Symmetry of the Matrix Operator L......Page 257
D.2 The Eigenfunctions Constitute a Set of Orthogonal Functions......Page 258
D.3 A detailed Derivation of Eq. (5.31) and Eq. (5.61)......Page 259
D.4 Simplification of the Expression for the Temperature Distribution (for Constant Wall Temperature)......Page 261
D.5 Simplification of the Expression for the Temperature Distribution (for Constant Wall Heat Flux)......Page 262
D.6 The Vector Norm......Page 265
References......Page 269
Index......Page 279

Citation preview

Bernhard Weigand Analytical Methods for Heat Transfer and Fluid Flow Problems

Bernhard Weigand

Analytical Methods for Heat Transfer and Fluid Flow Problems With 80 Figures

1 23

Professor Dr.-Ing. Bernhard Weigand Universität Stuttgart Institut für Thermodynamik der Luft- und Raumfahrt Pfaffenwaldring 31 70569 Stuttgart www.uni-stuttgart.de/itlr/ [email protected]

isbn 3-540-22247-2 Springer Berlin Heidelberg New York

Library of Congress Control Number: 2004107594

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitations, broadcasting, reproduction on microfi lm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer. Part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: data delivered by author Cover design: deblik Berlin Printed on acid free paper 62/3020/m - 5 4 3 2 1 0

For Irmi, my wife

Preface

Partial differential equations are the basis for nearly all technical processes in heat transfer and fluid mechanics. In my lectures over the past seven years I became aware of the fact that a lot of the students studying mechanical or aerospace engineering and also a lot of the engineers in industry today focus more and more on numerical methods for solving these partial differential equations. Analytical methods, taught in undergraduate mathematics, in thermodynamics and fluid mechanics, are quickly discarded, because most people believe that almost all problems, appearing in reall applications, can easily be solved by numerical methods. In addition, most of the examples shown in basic lectures are so simple that the students develop the impression that analytical methods are inappropriate for more complicated realistic technicall problems. It was exactly the above described mind set, which inspired me several years ago to give lectures on analytical methods for heat and mass transfer problems. The basic idea of these lectures is to show some selected analytical methods and to explain their application to more complicated problems, which are technically relevant. Of course, this means that some of the standard analytical methods, might not be discussed in these lectures and are also not present in this book (for example integral transforms). On the other hand, it can be shown that the analytical methods discussed here are applicable to interesting problems and the student or engineer learns how to solve useful technical problems analytically. This means that the main intent of this book is to show the usefulness of analytical methods, in a world, which focuses more and more on numerical methods. Of course, there is no doubt that the knowledge of numerical solution methods is very important and there is a big chance in using numerical tools to gain inside into flow physics and heat transfer characteristics. However, numerical methods are always dependent on grid quality and grid size and also on a lot of implementation features. Analytical methods can be used to validate and improve numerical methods. So the engineer might simplify a problem up to the extent that he can obtain an analytical solution. This analytical solution might be used afterwards to check and to improve the numerical solution for the full problem without any simplifications. This book has been written for graduate students and engineers. The mathematics needed to understand the solution approach is developed mostly during the actual solution of the problem under consideration. This means that the book includes only few proofs. The reader is referred in these situations to other books for these more basic mathematical considerations. This approach has been taken in

VIII

Preface

order to keep the focus of the book on the solution method itself and not to disrupt the analysis of the technical problem. The book is structured in six chapters. Chapter 1 provides a short introduction to the topic. Chapter 2 provides an introduction to the solution of linear partial differential equations. After discussing the classification and the character of the solutions of second-order partial differential equations, the method of separation of variables is discussed in detail. Chapter 3 is concerned with the solution of thermal entrance problems for pipe and channel flows. This means that solutions of the energy equation are considered for a hydrodynamically fully developed velocity profile. These problems lead to the solution of Sturm-Liouville eigenvalue problems, which are discussed for laminar and turbulent flows and for different wall boundary conditions. For the problems considered in Chap. 3, axial heat conduction within the flow can be ignored, because the Peclet number in the flow is sufficiently large. This means that the problems under consideration are parabolic in nature. Because this sort of eigenvalue problems normally cannot be solved analytically, a numerical procedure is discussed on how to solve them. In Appendix C, this numerical solution method is explained in detail and the reader is provided links to an internet page, containing several source codes and executables. Chapter 4 explains analytical solution methods for Sturm-Liouville eigenvalue problems for large eigenvalues. Here the focus is to explain an asymptotic analysis for a complicated problem, which is technically relevant. This chapter also provides comparisons between numerically and analytically predicted eigenvalues and constants. These comparisons show the usefulness of the analytical solution. Furthermore, it is explained how the method can be used for related problems. In Chapter 5 the heat transfer in pipe and channel flows for small Peclet numbers is considered. In contrast to the problems discussed in Chaps. 3-4, the axial heat conduction in the fluid cannot be ignored. This leads to elliptic problems. A method is presented, which gives rise to solutions, which are as simple as the ones presented in Chap. 3. The extension of this method to more complicated problems, for example for the heat transfer in hydrodynamically fully developed duct flows with a heated zone of finite length, is also explained. Chapter 6 is devoted finally to the solution of nonlinear partial differential equations. The idea behind this chapter was to provide a short overview about different solution methods for nonlinear partial differential equations. However, the main focus is on the derivation of similarity solutions. Here, different solution methods are explained. These are the method of dimensional analysis, grouptheory methods and the method of the free parameter. The methods are demonstrated for a simple heat conduction problem as well as for complicated boundary layer problems. Many people helped me in all phases of the preparation of this book. I am very grateful for many helpful discussions with my colleague Prof. Jens von Wolfersdorf concerning all aspects of the analytical solution methods. I also thank very much Martin Stricker and Marco Schüler who helped me with the figures. Many thanks also go to Dr. Grazia Lamanna for the helpful discussions and her support

Preface

IX

finishing this book. Also, I would like to thank Karl Straub very much for reading the manuscript. I kindly acknowledge the permission of the ASME for reprinting the Figs. 1.21.3 and of ELSEVIER for reprinting the Figs. 3.7, 3.13-3.15, 5.1, 5.9-5.13, 5.185.19, 5.25 in this book. I also kindly acknowledge the permission of the Council of Mechanical Engineers (IMECHE) for reprinting Fig. 3.12, of KLUWER for reprinting the Figs. 5.3-5.4, B2-B7 and of SPRINGER for reprinting the Figs. 3.8, 3.9, 3.17-3.19 in this book. In addition, I kindly acknowledge the permission of Prof. E. Papoutsakis for reprinting the Figs. 5.3-5.4 and of Prof. Osterkamp, Dr. Zhang and Dr. Gosink for reprinting Fig. 1.4 in this book. The reference of the paper, where the figures have originally been published, is always included in the individual figure legend. Finally, I am very grateful for the very good cooperation with Springer Press during the preparation of this manuscript. Here I would like to thank in particular Mrs. Maas, Mrs. Jantzen and Dr. Merkle for their support.

Filderstadt, April 2004

Bernhard Weigand

Contents

List of Symbols.................................................................................................... XV List of Copyrighted Figures .............................................................................XIX 1 Introduction......................................................................................................... 1 Problems......................................................................................................... 9 2 Linear Partial Differential Equations ............................................................. 11 2.1 Classification of Second-Order Partial Differential Equations .................. 11 2.2 Character of the Solutions for the Partial Differential Equations .............. 19 2.2.1 Parabolic Second-Order Equations ..................................................... 19 2.2.2 Elliptic Second-Order Equations ........................................................ 21 2.2.3 Hyperbolic Second-Order Equations .................................................. 23 2.3 Separation of Variables .............................................................................. 24 2.3.1 One-Dimensional Transient Heat Conduction in a Flat Plate............. 24 2.3.2 Steady-State Heat Conduction in a Rectangular Plate........................ 31 2.3.3 Separation of Variables for the General Case of a Linear SecondOrder Partial Differential Equation.............................................................. 37 Problems....................................................................................................... 39 3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems) ................ 43 3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature ...................................................................................................... 43 3.1.1 Velocity Distribution of Hydrodynamically Fully Developed Pipe and Channel Flows.............................................................................................. 44 3.1.2 Thermal Entrance Solutions for Constant Wall Temperature ............ 46 Properties of the Sturm-Liouville System.................................................... 50 3.2 Thermal Entrance Solutions for an Arbitrary Wall Temperature Distribution ....................................................................................................... 66 3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux ................................................................................................................... 69 3.3.1 Velocity Distribution for the Hydrodynamically Fully Developed Flow in an Axial Rotating Pipe.................................................................... 72 3.3.2 Thermal Entrance Solution for Constant Wall Heat Flux................... 75 Problems....................................................................................................... 83

XII

Contents

4 Analytical Solutions for Sturm - Liouville Systems with Large Eigenvalues ........................................................................................................... 87 4.1 Heat Transfer in Turbulent Pipe Flow with Constant Wall Temperature 103 4.2 Heat Transfer in an Axially Rotating Pipe with Constant Wall Temperature.................................................................................................... 109 4.3 Asymptotic Expressions for other Thermal Boundary Conditions .......... 113 Problems..................................................................................................... 115 5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems).............................................................................................. 117 5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0 ........ 120 5.1.1 Heat Transfer in Laminar Pipe and Channel Flows for Small Peclet Numbers ..................................................................................................... 131 5.1.2 Heat Transfer in Turbulent Pipe and Channel Flows for Small Peclet Numbers ..................................................................................................... 138 5.2 Heat transfer for Constant Wall Heat Flux for x d 0 and x > 0 ................ 142 5.2.1 Heat Transfer in Laminar Pipe and Channel Flows for Small Peclet Numbers ..................................................................................................... 147 5.2.2 Heat Transfer in Turbulent Pipe and Channel Flows for Small Peclet Numbers ..................................................................................................... 150 5.3 Results for Heating Sections with a Finite Length................................... 153 5.3.1 Piecewise Constant Wall Temperature ............................................. 154 5.3.1 Piecewise Constant Wall Heat Flux.................................................. 157 5.4 Application of the Solution Method to Related Problems ....................... 159 Problems..................................................................................................... 161 6 Nonlinear Partial Differential Equations ..................................................... 165 6.1 The Method of Separation of Variables ................................................... 165 6.2 Transformations Resulting in Linear Partial Differential Equations ....... 172 6.3 Functional Relations Between Dependent Variables ............................... 174 6.3.1 Incompressible Flow over a Heated Flat Plate ................................. 174 6.3.2 Compressible Flow over a Flat, Heated Plate................................... 177 6.4 Similarity Solutions .................................................................................. 179 6.4.1 Similarity Solutions for a Transient Heat Conduction Problem....... 179 6.4.2 Similarity Solutions of the Boundary Layer Equations for Laminar Free Convection Flow on a Vertical Flat Plate.......................................... 186 6.4.3 Similarity Solutions of the Compressible Boundary Layer Equations.................................................................................................... 192 Problems..................................................................................................... 198 Appendix A: The Fully Developed Velocity Profile for Turbulent Duct Flows .................................................................................................................... 203 Appendix B: The Fully Developed Velocity Profile in an Axially Rotating Pipe ...................................................................................................................... 215

Contents

XIII

Appendix C: A Numerical Solution Method for Eigenvalue Problems ........ 227 C.1 Numerical Tools....................................................................................... 229 Appendix D: Detailed Derivation of Certain Properties of the Method for Solving the Extended Graetz Problems............................................................ 235 D.1 Symmetry of the Matrix Operator L ..................................................... 235  D.2 The Eigenfunctions Constitute a Set of Orthogonal Functions.............. 236 D.3 A detailed Derivation of Eq. (5.31) and Eq. (5.61) ................................ 237 D.4 Simplification of the Expression for the Temperature Distribution (for Constant Wall Temperature) .......................................................................... 239 D.5 Simplification of the Expression for the Temperature Distribution (for Constant Wall Heat Flux) ............................................................................... 240 & 2 D.6 The Vector Norm ) j .......................................................................... 243 References ........................................................................................................... 247 Index .................................................................................................................... 257

List of Symbols

a a1 a2 A Aj Bi c cf cp C D E F Fx , Fy , Fz

[m2/s] [-] [m2] [-] [-] [m] [-] [J/(kg K)] [-] [m] [-] [-] [N]

heat diffusivity functions flow area constants Biot number velocity of sound friction factor specific heat at constant pressure Chapman-Rubesin parameter hydraulic diameter dimensionless energy flow flow index (0 for planar channel, 1 for pipe) forces

G h h Ji ( )

[-] [W/(m2K)] [m] [-]

function heat transfer coefficient half channel height of a planar channel Bessel function of order i

k k K L Lth L[ ]  l M[ ] Ma ∞

[W/(m K)] [-] [W/m3] [m] [m] [-]

thermal conductivity transformed eigenvalue sink intensity length scale (h for planar channel, R for pipe) thermal entrance length matrix operator

[m] [-] [-]

mixing length operator Mach number

Nu L

[-]

Nusselt number based on L

Nu ∞ N n p Pr Prrt

[-] [-] [m] [Pa] [-] [-]

Nusselt number for the fully developed flow rotation rate coordinate orthogonal to the flow direction pressure Prandtl number turbulent Prandtl number

XVI

List of Symbols

PeeL Pet R R ReeL Ri t T T′ TW Tb uτ

[-] [-] [m] [J/(kg K)] [-] [-] [s] [K] [K] [K] [K] [m/s]

Peclet number based on L turbulent Peclet number pipe radius gas constant Reynolds number based on L Richardson number time temperature temperature fluctuation wall temperature bulk–temperature shear velocity

U u u, v, w u ′, v′, w′

[m] [m/s] [m/s] [m/s] [-] [m] [-] [-]

wetted perimeter mean velocity velocity components fluctuating velocity components dimensionless velocity gradient at the wall coordinates wall coordinate

V x, y, z y+ Z

modified rotation parameter

Greek letter symbols

β β1 β 2 Γ εm ε hx , ε hy , ε hr

[1/K]

volumetric coefficient of expansion

[-] [-] [m2/s]

constants gamma function eddy diffusivity for momentum transfer

[m2/s]

eddy diffusivity for heat transfer

ξ ,η η ϑ Θ Θb λj

[-] [-] [-] [-] [-]

characteristic coordinates similarity variable transformed eigenfunction dimensionless temperature dimensionless bulk-temperature

[-]

eigenvalue

μ ν ρ τ φ

[kg/(m s)] [m2/s] [kg/m3] [-] [-]

dynamic viscosity kinematic viscosity density dimensionless time enthalpy function

List of Symbols

Φj

[-]

Φ Dis

[1/s ]

dissipation function

Φ Ψ ω

[m /s ] [m2/s] [1/s]

velocity potential stream function angular velocity

eigenfunctions 2

2

Subscripts

0 C ∞ , +

refers to inlet conditions centerline of the duct free stream conditions dimensionless quantities

Definition of non-dimensional Numbers

Bi =

hD k

Biot number

C=

ρμ ρ μ∞

Chapman-Rubesin parameter

N=

Reϕ

Rotation rate

Re D

∂T L hL ∂n W Nu L = = k TW Tb −

u∞

Ma ∞ =

uL a

Pe L

Pr =

κ p/ρ

μcp

Re L =

k

Re L Pr

=

uL

ν

ν a

Nusselt number

Mach number

Peclet number

Prandtl number

Reynolds number

XVII

XVIII

List of Symbols

Reϕ =

wW D

Reτ =

uτ L

Rotational Reynolds number

ν

Shear stress Reynolds number

ν

w ∂ ( ) r ∂r Ri = 2 § ∂u · § ∂ § w · · ¨ ¸ + r ¨ ¸¸ © ∂rr ¹ © ∂ r © r ¹ ¹ 2

Z N

Re D 2 Reτ

N/

cf 8

2

Richardson number

Modified rotation parameter

List of Copyrighted Figures

I kindly acknowledge the permission of the ASME for reprinting the Figs. 1.2-1.3, which have been published first in Burow P, Weigand B (1990) One-dimensional heat conduction in a semi-infinite solid with the surface temperature a harmonic function of time: A simple approximate solution for the transient behavior. Journal of Heat Transfer 112: 1076 – 1079 (Fig. 1). I kindly acknowledge the permission of ELSEVIER for reprinting the Figs. 3.7, 5.9 and 5.18 which have been published first in Weigand B, Ferguson JR, Crawford ME (1997a) An extended Kays and Cawford turbulent Prandtl number model, Int. J. Heat Mass Transfer, 40: 4191- 4196 (Figs. 3-4). In addition, I kindly acknowledge the permission of ELSEVIER for reprinting the Figs. 3.13-3.15, which have been first published in Reich G, Beer H (1989) Fluid flow and heat transfer in an axially rotating pipe-I. Effect of rotation on turbulent pipe flow, Int. J. Heat Mass Transfer 32: 551-562 (Figs. 1, 3-4) and I kindly acknowledge the permission of ELSEVIER for reprinting the Figs. 5.1, 5.10 - 5.11, which have been first published in Weigand B (1996) An exact analytical solution for the extended turbulent Graetz problem with Dirichlet wall boundary conditions for pipe and channel flows, Int. J. Heat Mass Transfer 39: 1625-1637 (Figs. 1, 5-6). In addition, I kindly acknowledge the permission of ELSEVIER for reprinting the Figs. 5.12-5.13, 5.19 which have been first published in Weigand B, Kanzamar M, Beer H (2001) The extended Graetz problem with piecewise constant wall heat flux for pipe and channel flows, Int. J. Heat Mass Transfer 44: 3941-3952 (Figs. 1, 9). I also kindly acknowledge the permission of the Council of Mechanical Engineers (IMECHE) for reprinting Fig. 3.12, which has been first published in White A (1964) Flow of a fluid in an axially rotating pipe, Journal Mechanical Engineering Science 6: 47-52 (Figs.10-11). I kindly acknowledge the permission of KLUWER for reprinting the Figs. 5.35.4, which have been first published in Papoutsakis E, Ramkrishna D, Lim H (1980) The extended Graetz problem with Dirichlet wall boundary conditions, Applied Scientific Research 36: 13-34 (Figs. 1-3) and I kindly acknowledge the permission of KLUWER for reprinting Figs. B2-B7, which have been first published in Weigand B, Beer H (1994) On the universality of the velocity profiles of a turbulent flow in an axially rotating pipe, Applied Scientific Research 52: 115132 (Figs. 5-7, 9-10, 12). I kindly acknowledge the permission of SPRINGER for reprinting the Figs. 3.8, 3.9, 3.17-3.19 which have been first been published in Weigand B, Beer H (1989) Wärmebertragung in einem axial rotierenden, durchströmten Rohr im Bereich des thermischen Einlaufs, Wärme- und Stoffübertragung 24: 191-202

XX

List of Copyrighted Figures

(Figs. 4 - 6, 9) and I kindly acknowledge the permission of SPRINGER for reprinting Fig. 5.25 which has been first published in Weigand B, Wrona F (2003) The extended Graetz problem with piecewise constant wall heat flux for laminar and turbulent flows inside concentric annuli, Heat and Mass Transfer 39: 313-320 (Fig. 2). In addition, I kindly acknowledge the permission of Prof. E. Papoutsakis for reprinting the Figs. 5.3-5.4, which have been first published in Papoutsakis E, Ramkrishna D, Lim H (1980) The extended Graetz problem with Dirichlet wall boundary conditions, Applied Scientific Research 36: 13-34 (Figs. 1-3). I also kindly acknowledge the permission of Prof. Osterkamp, Dr. Zhang and Dr. Gosink for reprinting Fig. 1.4, which has been first published in Zhang T, Osterkamp TE, Gosink JP (1991) A model for the thermal regime of Permafrost within the depth of annual temperature variations. Proc. 3rd Int. Symp. on Therm. Eng. Sci. for Cold Regions, Fairbanks, Alaska, USA, pp. 341 – 347 (Fig. 3).

1 Introduction

Fluid flow and heat transfer problems are present in all of our daily life. For example, if we walk along a river and look at the water flowing with high speed over the river-bed, we actually observe a fluid mechanics problem. If we put some sugar into our coffee and stir it, we are faced with a complicated heat and mass transfer problem. In particular, convective heat transfer problems are present everywhere in our world. Most of the problems encountered in fluid mechanics or heat transfer are described by partial differential equations. One good example of such equations are the Navier-Stokes equations and the energy equation for an incompressible flow with constant fluid properties. If we consider a threedimensional, steady-state problem, these equations read

§

§

2

2 2 u u u· + 2 + 2¸ 2 ∂ x ∂ y ∂ z © ¹

§

2

v

u ∂y

w

u· ¸ ∂z ¹

Fx

∂pp ∂x

μ¨

v

v ∂y

w

v· ¸ ∂z ¹

Fy

∂pp ∂y

μ¨

w w w· v w ¸ ∂y ∂z ¹ © ∂x

Fz

∂pp ∂z

μ¨

u ∂ © x

ρ ¨u §

v © ∂x

ρ ¨u

2 2 v v v· + + ¸ 2 2 ∂y ∂z 2 ¹ © ∂x

(1.2)

2 2 w w w· + + ¸ 2 2 ∂y ∂z 2 ¹ © ∂x

(1.3)

2 2 § 2T T T T· T T· +v +w ¸ = μ Φ Dis + k ¨ 2 + 2 + 2 ¸ ∂ x ∂ y ∂ z ∂ x ∂ y ∂ z © ¹ © ¹

(1.4)

§

ρ ¨u

§

(1.1)

2

§

ρc ¨ u

where Φ Dis denotes the dissipation function in the energy equation. This function is given by 2 ª§ ∂uu ·2 § v · 2 § w ·2 º § v u· Φ Dis = 2 «¨ ¸ + ¨ ¸ + ¨ + + » ¨ ¸ + z ¸ » © ∂x ∂y ¹ «¬© ∂xx © y¹ ¼ 2

§ w v · § ∂u ∂w · +¨ + ¸ +¨ + ¸ z x¹ © ∂y ∂z ¹

(1.5)

2

This set of equations is closed by adding the incompressible mass continuity equation

2

1 Introduction

∂u ∂v ∂w + + =0 ∂x ∂y ∂z

(1.6)

In the above equations, x, y, z denote the cartesian coordinates, p indicates the pressure, T the temperature, Fx, Fy and Fz are forces, and u, v, w are the velocity components in the x, y and z direction, respectively. From the above equations it can be seen that fluid flow and heat transfer problems are described by a set of partial differential equations. In general, these complicated differential equations can only be solved numerically. However, analytical solutions for fluid mechanics or heat transfer problems can still play an important role in science and in engineering, even in the current age of supercomputers. This is because analytical solutions have the big advantage of showing directly which parameters influence the solution. This is illustrated by the following short example (see Fig. 1.1). We are interested in the answer to the question on how a periodic change of the surface temperature of the earth will influence the temperature in the soil. Here, we are primarily interested in the behavior for increasing x, i.e. the distance from the surface, and growing values of time. Since we are only interested in the temperature change in the radial direction, we can approximate the earth as a semi-infinite body. In addition, we might use a cartesian coordinate system to describe this problem, because the radius of curvature of the earth is very large compared to all other dimensions. T

T1 cos(ωtt

)

x semi-infinite body

Fig. 1.1: Semi-infinite body with a periodically changing surface temperature

If we write down the energy equation for this problem and if we consider the physical properties of the body to be constant, the energy equation can be simplified to

ρc

∂T ∂t

2 2 § 2T T T· k¨ 2 + 2 + 2 ¸ y z ¹ © x

(1.7)

Furthermore, the heat conduction in the y- and z- direction can be neglected compared to the heat conduction in the x-direction. If we do so, we obtain

∂T ∂ 2T =a 2 ; ∂t ∂x

a=

k ρc

The boundary conditions for this problem are given by

(1.8)

1 Introduction

T1 cos (

x = 0: T

t =0: T

)

T0 ;

3

(1.9) (1.10)

0

where T0 is the constant initial temperature of the solid body and ω and ε are given constants. The solution of this problem is (see Carslaw and Jaeger (1992)) Θ = e −η cos (

2

)

η / 2τ

³

π

0

η2 2μ 2

ª cos «τ ¬

º

ε»e ¼

μ2

dμ

(1.11)

with the dimensionless quantities Θ=

T T0 , τ T1

ωt ,

η=x

ω

(1.12)

2a

From the above given analytical solution (1.11), one sees very clearly that the solution consists of two parts. The second part of Eq. (1.11) determines the behavior of the solution for short times

ΘT =

η / 2τ

2

π

³ 0

º η2 ε»e 2 2μ ¼

ª cos «τ ¬

μ2

dμ

(1.13)

For a fixed value of x ( η = const.), this part of the solution tends to zero with increasing time, as it can be seen from the upper boundary of the integral. The first part of the solution (1.11) is a periodic part, which is multiplied by an exp-function Θ P = e−η cos((

)

(1.14)

From this part of the solution, we can see that the amplitude of the oscillation of the surface temperature decreases with increasing values of η. Additionally, one can notice that the maximum of the oscillation appears at different depths of the material with a time delay. Furthermore, it can be seen that the solution for Θ only depends on the following dimensionless quantities:

τ η=x

ωt ω

(1.15) (1.16)

2a

In this context, it is of special interest to have a closer look on the variable η. If ω / ( ) is large (which means: ω is large or a k / ( ) is very small, for example because of a low heat conducting material), η will be large even for small values of x. This means that the temperature wave will be damped out very fast. The behavior of the solution (1.11) is demonstrated for one example. We consider the case ε = 0. The complete solution is depicted in Fig. 1.2. It can be seen

4

1 Introduction

from Fig. 1.2 that the maximum temperature appears at different times for increasing values of η. In addition, it is clear from Fig. 1.2 that the solution satisfies the initial condition that Θ = 0 for τ = 0 inside the solid body (i.e. η > 0). Figure 1.3 shows only the first part of the solution Θ P . As stated before, it can be seen that Θ P is a very good approximation of the complete solution of the problem for larger times. For τ > 0.1 the complete solution is nicely approximated.

Fig. 1.2: Temperature distribution for ε = 0 as a function of η and τ (Burow and Weigand (1990))

Fig. 1.3: Temperature distribution Θ P for ε = 0 as a function of η and τ (Burow and Weigand (1990))

1 Introduction

5

This also shows that, for some problems, it might suffice to determine the solution Θ P and not the full solution of the problem. This sort of solution can be determined very easily also for more complicated variations of the surface temperature (see for example Myers (1987) or Carslaw and Jaeger (1992)). One example is depicted in Fig. 1.4, which shows the annual oscillation of the temperature in the soil in Alaska at various depths. The figure shows nicely how the minimum temperature at a certain depth in the soil gets shifted to different times. In addition it can be seen how the amplitude of the temperature oscillation decreases with increasing depth. This leads to the interesting fact, that at certain times in the year the temperature stratification in the soil can get reversed.

Fig. 1.4: Annual oscillation of the soil temperature in Alaska at different depths (Zhang et al. (1991))

The examples discussed above show very clearly where the strength of the analytical methods lay. The analytical solution shows clearly the dependence of the solution on the dimensional quantities. Furthermore, short time effects or long time behavior are clearly visible. Analytical solutions might therefore be derived for strongly simplified problems, before a numerical solution is carried out. The simplified model can then be used to understand the basic physical phenomena and behavior of the problem. Additionally, analytical solutions can be used for checking the numerical calculations and proving that all the settings, e.g. grid quality, meshing technique, numerical scheme and so on, are adequate for the considered problem. This is especially important for nonlinear problems, where grid size and grid quality might play a very important role for the final accuracy obtained by the numerical method. In a next step, we want to address the question on how to classify the partial differential equations (PDE). In general, partial differential equations are classi-

6

1 Introduction

fied by their order and by the fact if they are linear, quasilinear or nonlinear (the classification of second order linear partial differential equations is discussed in detail in Chap. 2). An example of a linear second-order partial differential equation is the energy equation describing the heat conduction within a solid body, which has constant physical properties:

ρc

∂T ∂t

(1.17)

2 2 § 2T T T· k¨ 2 + 2 + 2 ¸ x y z © ¹

Another example of a linear second-order partial differential equation is the potential flow equation. Here we are considering an incompressible, irrotational flow with velocity components u and v. The velocity components are related to the velocity potential Φ by u

∂Φ , ∂x

v=

∂Φ ∂y

(1.18)

From mass continuity, one obtains the following partial differential equation for the function Φ

∂ 2Φ ∂ 2Φ + =0 ∂x 2 ∂y 2

(1.19)

In contrast to the above given equations, the partial differential equation, which describes the flow of a compressible inviscid flow, is a nonlinear second-order partial differential equation 2 § § ∂Φ ·2 · 2 · ∂Φ ∂Φ ∂ 2 Φ § § 2 ∂ Φ − c + 2 ¨¨ ¨ ¸ ¸ ¸ 2 ∂x ∂yy ∂xx∂yy ¨© © y ¹ © © ∂xx ¹ ¹ ∂x

2

· ∂ 2Φ

¸ ∂yy 2 ¹

0

(1.20)

where Φ denotes in Eqs. (1.18–1.20) the velocity potential and c is the speed of sound. Another example of a nonlinear partial differential equation is the energy equation describing the one-dimensional transient heat conduction within a solid body whose thermal conductivity is a function of temperature. Additionally, an energy source term is incorporated in the material:

ρc

∂T ∂t

§ ∂ § ∂T · · k (T ) ¨ ¸ ¸ qi ( ∂x x ∂xx ¹ ¹ © ©

)

(1.21)

Summarizing, one can say that the order of a partial differential equation is given by the highest-ordered partial derivative appearing in the equation. A partial differential equation is called linear if it is linear in the unknown function and all its derivatives, i.e. the multiplying coefficients depend only on the independent variables. If the equation is linear in the highest-order derivative, it is called quasilinear. The Navier-Stokes equations are a good example of quasilinear equations. If the nonlinearity appears also in the highest-order derivative, it is called a nonlinear

1 Introduction

7

partial differential equation. Finally, it should be noted that the equation is called nonhomegeneous, if it contains one term which is only a function of the independent variables, otherwise it is called homogeneous. Equation (1.21) is a nonhomogeneous equation, because it contains a heat source which is a function of x. On the other hand Eqs. (1.17, 1.19, 1.20) are homogeneous equations. The striking advantage in dealing with linear partial differential equations is that the final solution can be constructed by using the superposition method. This means that we can superimpose a number of solutions of much simpler problems to finally obtain the solution of a complicated problem. This is illustrated by the following simple example. We consider the heat conduction within a flat plate which has the dimensions L in the x and y direction. The region contains a heat source and the four boundaries are kept at different temperatures. The physical properties of the material are considered constant. The problem is then described by the following sketch and equations:

y L Θ = g4 ( y )

Θ = g3 ( x )

∂ 2Θ ∂ 2Θ + = f ( x, y ) ∂ x2 ∂ y2 Θ( , ) g1 ( ) Θ = g2 ( y ) ΔΘ = f ( x , y ) Θ ( , ) g2 ( ) Θ ( , ) g3 ( ) Θ ( , y ) g4 ( y ) x Θ = g1 ( x ) L

Since the problem under consideration is described by a second-order linear partial differential equation, the solution can be obtained by superimposing the solutions of the following five simpler problems:

y L Θ1 = 0

Θ1 = 0

ΔΘ1 = f x y) y

Θ1 = 0

ΔΘ 1 = f ( x , y )

Θ1 = 0

+

L

x

Θ1 ( x ,0) 0 Θ1 ( L, y) = 0 Θ1 ( x , L) = 0 Θ1 ( 0, y ) 0

8

1 Introduction

y

Θ2 = 0

L Θ2 = 0

ΔΘ 2 = 0

Θ2 = 0

ΔΘ 2 = 0

Θ 2 ( ,0) = 1 ( ) Θ 2 ( , y) = 0 Θ2 ( , ) = 0 Θ 2 (0, y ) 0

Θ 2 = g1 ( x )

x

L

+ y

Θ3 = 0

ΔΘ 3 = 0

L Θ3 = 0

ΔΘ 3 = 0

Θ 3 = g2 ( y)

Θ 3 ( ,0)

0

Θ3 ( , ) =

2

( y)

Θ3 ( , ) = 0 Θ 3 (0, ) Θ3 = 0

L

0

x

+ y

Θ 4 = g3( x)

ΔΘ 4 = 0

L Θ4 = 0

ΔΘ 4 = 0

Θ4 = 0

Θ 4 ( ,0) 0 Θ4 ( , ) = 0 Θ4 ( , ) = 3 ( ) Θ 4 ( 0, )

Θ

+

4

= 0

L

x

0

1 Introduction

y L Θ 5 = g4 ( y)

Θ5 = 0

ΔΘ 5 = 0 Θ5 ( ,0)

Θ5 = 0

ΔΘ 5 = 0

5

4

( y)

x

L

= 0

0

Θ5 ( , y) = 0 Θ5 ( , ) = 0 Θ5 (0, ) =

Θ

9

After solving the above given individual problems, we know the solutions Θ1 2 3 4 and d Θ5 . The solution of the problem can then be obtained by simply adding all the solutions of the five sub-problems 5

Θ = ¦ Θi

(1.22)

i =1

This can be visualized by looking at the individual equations and the related boundary conditions. Adding up the five individual partial differential equations results in ∂ 2 (Θ1 + Θ2 + Θ3 + Θ4 + Θ5 ) ∂ 2 (Θ1 + Θ2 + Θ3 + Θ4 + Θ5 ) + = f( , ) ∂x 2 ∂y 2

(1.23)

By adding up the boundary conditions of the individual problems, the boundary conditions of the original problem will be obtained. This example shows clearly how powerful the superposition method is. Furthermore, the superposition method makes linear partial differential equations so much easier to solve than nonlinear partial differential equations, where the superposition approach fails. Problems

1-1. State for each of the following partial differential equations if it is linear, quasilinear or nonlinear. In addition, state if the equation is homogeneous or nonhomogeneous and give its order:

ρc

∂T ∂xx

k

∂ 2T ∂yy 2

∂ 2T ∂ 2T + = f ∂x 2 ∂y 2

(

)

10

1 Introduction

∂ 4u ∂ 4u ∂ 4u + 4 +3 2 2 4 ∂x ∂y ∂x ∂y 2

∂ 2u ln u ∂xx 2

∂u ∂xx

3(

5 xy

)

∂ 2u ∂u +u = ey ∂x 2 ∂x 1-2. Consider the steady-state temperature distribution in a quadratic plate. The problem is given by

∂2Θ ∂2Θ + =0 ∂x 2 ∂y 2 with the boundary conditions Θ( Θ( Θ( Θ(

)=0 ) ( )=0 ) (

) )

a.) Use the superposition method to solve this problem. Split therefore the problem in two different problems (one which includes only the nonhomogeneous boundary condition Θ1 ( ) ( ) and one including the condition with Θ 2 ( ) ( ) ). b.) By adding the equations and boundary conditions of the two problems Θ1 + Θ 2 , show that the original problem is obtained. c.) Show by inserting, that the two problems have the solutions: Θ1 = Θ2 = d.)

sinh (

) sin ( sinh ( )

sinh (

sinh (

)

)

sin (

) )

Show by insertion that Θ = Θ1 + Θ2 satisfies the original problem.

2 Linear Partial Differential Equations

Several important heat and fluid flow processes in technical applications and in nature can approximately be described by linear partial differential equations. As stated in the previous chapter, linear partial differential equations are normally much simpler to solve than nonlinear partial differential equations. In addition, a large body of literature exists on how to solve linear PDEs. The following four chapters focus on the solution of linear partial differential equations. Chapter 2 is concerned with the classification of second order partial differential equations and presents a short introduction into the method of separation of variables. Chapter 3 focuses on convective heat transfer in laminar and turbulent pipe and channel flows. Here parabolic problems are considered and the general eigenvalue problems, associated with these equations, are explained. Chapter 4 discusses some specific methods for the analytical solution of eigenvalue problems, in the case of large eigenvalues. Finally, Chap. 5 deals with convective heat transfer problems in laminar or turbulent pipe and channel flows for low Peclet numbers (liquid metals). For this type of applications, the axial heat conduction within the flow can no longer be ignored and the resulting energy equation remains elliptic in nature. This has strong implications on the solution method for the energy equation.

2.1 Classification of Second-Order Partial Differential Equations In the following, we are concerned with a linear second order partial differential equation which depends on the two independent variables x and y. The most general form of the homogeneous equation is given by

∂ 2u ∂2u ∂ 2u 2 ( , ) ( ) 2+ 2 ∂x ∂x∂y ∂y ∂u ∂u +D D ( x y) E ( x, y ) F ( x, y ) u 0 ∂x ∂y A( x y )

(2.1)

where A,...,F F are constants or functions of x and y and are sufficiently differentiable in the domain of interest. The form of Eq. (2.1) resembles the quadratic equation of a conic sections in analytical geometry. The equation

12

2 Linear Partial Differential Equations

a x2

2b xy c y 2 d x e y

f

0

(2.2)

represents an ellipse, parabola or hyperbola depending on whether ac-b2 0, respectively. The classification of the second-order partial differential equation is based on the fact that Eq. (2.1) can be transformed into a standard form. This is very similar to the treatment of the quadratic equation (2.2) of conic sections in analytical geometry. We distinguish the following different cases (for the point x0, y0 under consideration): 1. B 2 ( x0 ,

0

)

( 0,

0

) ( 0,

0

)

0

(2.3)

hyperbolic type. There exist two real characteristic curves. 2. B 2 ( x0 ,

0

)

( 0,

0

) ( 0,

0

)

0

(2.4)

parabolic type. There exists one real characteristic curve. 3. B 2 ( x0 ,

0

)

( 0,

0

) ( 0,

0

)

0

(2.5)

elliptic type. The two characteristic curves are conjugate complex. Each of these equations can be transformed into its standard form, if we introduce the following new coordinates into Eq. (2.1)

ξ ξ( , ) η η( , )

(2.6)

Is the equation of hyperbolic type, the standard form is given by § ∂ 2u u u· = H1 ¨ ξ , η , u , , ¸ ∂ξ∂η ∂ξ ∂η ¹ ©

(2.7)

If one introduces the new coordinates

ξ ξ +η η ξ −η

(2.8)

we obtain an alternative standard form of the hyperbolic equation § ∂ 2u ∂ 2u u u· − 2 = H 2 ¨ ξ ,η , u , , ¸ 2 ∂ξ ∂η ¹ ∂ξ η © If the equation is of parabolic type, then the standard form is given by

(2.9)

2.1 Classification of Second-Order Partial Differential Equations

§ ∂ 2u u u· = H 3 ¨ ξ ,η , u , , ¸ ∂ ξ ∂η ¹ ∂ξ 2 ©

13

(2.10)

Finally, if the equation is of elliptic type, the standard form of the equation is given by

§ ∂ 2 u ∂ 2u u u· + 2 = H 4 ¨ ξ ,η ,,uu, , ¸ 2  ∂η ∂ξ ξ ∂η ¹ ©

(2.11)

where the new coordinates ξ ,η are defined by

1 ( 2 1 η ( 2i

ξ

(2.12)

) )

In order to obtain one of the above standard or canonical forms of the equation, we need to perform the coordinate transform given by Eq. (2.6). Since we want the transformed equation to be equivalent to the original equation, we assume that ξ and η are twice continuously differentiable and we insist that the Jacobian § ∂ξ ¨ ∂xx Jacobian = ¨ ¨ η ¨ © ∂xx

ξ·

y ¸ ∂ξ η ξ η ¸= − ≠0 η ¸ ∂x ∂y ∂y ∂x ¸ y¹

(2.13)

in the region under consideration. By assuming that Eq. (2.13) holds, we have always a unique transformation between x,y and ξ,η. Use of the chain rule gives: § ∂u · § ∂ξ · § ∂u · § ∂η · ∂u · § ∂u +¨ ¸ ¨ ¸ =¨ ¸ ∂ x ξ η © x ¹y η ξ ¨© x ¸¹ y © ¹y

(2.14)

§ u· § u· § ξ · § u · § η· ¨ ¸ = ¨ ¸ ¨ ¸ +¨ ¸ ¨ ¸ © ∂y ¹ x © ∂ξ ¹η © ∂y ¹ x © ∂η ¹ξ © ∂y ¹ x and also 2 2 ∂ 2 u ∂ 2 u § ∂ξ · u ξ η u η· = + 2 + + ¨ ¸ ∂x 2 ∂ξ 2 x ξ η x x η 2 ¨© ∂xx ¸¹ 2

+

∂u ∂ 2ξ u 2η + ∂ξ x 2 η ∂xx 2

2

(2.15)

14

2 Linear Partial Differential Equations

2 ∂ 2u ∂ 2 u ∂ξ ξ u § ξ η ξ η· = 2 + + ¨ ¸+ ∂x∂y ξ ∂x ∂y ∂ξ∂η © ∂x ∂y ∂y ∂x ¹ ∂ 2 u ∂η η u 2ξ u 2η + 2 + + ∂η ∂x ∂y ∂ξ ∂x∂y ∂η ∂x∂y

2

2

2 2 ∂ 2u ∂ 2u § ξ · u ξ η u η· = 2 ¨ ¸ +2 + 2¨ ¸ + 2 ∂y ∂ξ © y ¹ ξ η y y η © ∂yy ¹ ∂u ∂ 2ξ u 2η + + 2 ∂ξ y η ∂yy 2

After inserting the above expressions into Eq. (2.1), one obtains ∂2u ∂ξ 2 ∂u ) ∂ξ

A(

)

D(

(

2

(

∂ 2u

,

)

)

∂u

ξ η

(

)

∂ 2u

η

2

+

(2.16)

0

η

with the coefficients A

B=A

§ ∂ξ · A¨ ¸ © ∂xx

2

22B B

ξ ξ ∂x ∂y

§ ξ· C¨ ¸ © ∂yy ¹

2

(2.17)

§ ξ η ∂ξ ∂η ξ η· ∂ξ ∂η + B¨ + ¸+C ∂x ∂x ∂y ∂y © ∂x ∂y ∂y ∂x ¹

C

§ ∂η · A¨ ¸ © ∂xx

D

A

∂ 2ξ ∂x 2

2B 2 B

E

A

∂ 2η ∂x 2

2B

2

2B

ξ

∂η ∂η ∂x ∂y

2

∂x∂y

ξ

§ η· C¨ ¸ © ∂yy ¹

2

C

∂y

2

∂ 2η ∂ 2η C 2 ∂x∂y ∂y

D

D

ξ ∂x ∂η ∂x

2

E

E

ξ ∂y ∂η ∂y

Now we need to specify our change of variables, expressed by Eq. (2.6), in order to obtain one of the previously given standard or canonical forms. For example, if we want to obtain the hyperbolic equation in the form of Eq. (2.7) we have to assume that A and C are equal to zero. Because these two expressions are identical

2.1 Classification of Second-Order Partial Differential Equations

15

if we exchange ξ and η, we can achieve the condition A = C = 0, only when ξ and η are both solutions of the following equation § ∂Ω · A¨ ¸ © ∂xx

2

· § ∂Ω · § 2B ¨ ¸ ¸¨ x © ∂y ∂y ¹

§ · ¨ ¸ © ∂y ¹

2

0 ; Ω=ξ or η

(2.18)

Along a curve Ω = const. one has d

∂Ω dx d ∂x

∂Ω dyy = 0 y

∂Ω ∂Ω / ∂x ∂y

dy dx

(2.19)

we obtain from Eqs. (2.18) and (2.19) the following ordinary differential equation § dyy · A¨ ¸ © dx ¹

2

(2.20)

§ dy · 2B ¨ ¸ © dx ¹

0

This differential equation can be solved for dy/dx and one gets the following two cases:

( (

dy = B + B2 dx dy = dx

AC

) )

A

(2.21)

A

These two equations are known as the characteristic equations. They prescribe the functional relationship between the families of curves in the xy-plane for which ξ = const. and η = const.. This means that a change of variables according to

ξ = f1 ( , ) η = f2 ( , )

(2.22)

will transform Eq. (2.1) into its standard form. From Eq. (2.21) it is apparent that there are three cases to be considered: Case 1: Hyperbolic equation (B2 – AC) > 0 The preceding analysis results in a canonical form for the hyperbolic equation. For this case we have two real characteristics, which can be obtained from the differential Eqs. (2.21). Case 2: Elliptic equation (B2 – AC) < 0 For this case, one obtains from Eq. (2.21) no real, but two conjugate complex solutions. Therefore, the elliptic equation has two conjugate complex characteristics. The elliptic case needs not to be recalculated again, because it can be deduced from the calculation of the hyperbolic case. This can be shown as follows: let us consider the canonical form

16

2 Linear Partial Differential Equations

§ ∂ 2u u u· = H1 ¨ ξ , η , u , , ¸ ∂ξ∂η ∂ξ ∂η ¹ ©

(2.7)

Now, we have two conjugate complex characteristics. Therefore, we can introduce into the above equation the coordinate transform given by Eq. (2.12) 1 ( 2 1 η ( 2i

ξ

(2.12)

) )

Use of the chain rule according to Eq. (2.15) gives 2 ∂ 2u ∂ 2 u ∂ξ ξ u § ξ η ξ η· = 2 + + ¨ ¸+   ∂ξ η η ξ¹ ξ ξ η ξ η© ξ η ∂ 2 u ∂η η u 2ξ u 2η + 2 + + ∂η ∂ξ η η ξ η ξ ξ η

=

2 ∂ 2u 1 ∂ 2u § 1 1 · ∂ 2 u 1 1 § 2u u· + − + + = + ¨ ¸ 2 2 2  ∂η 2 ¹ ∂ξ 4 ξ η © 4i 4i ¹ ∂η 4 4 © ∂ξ

and Eq. (2.7) transforms into the standard form for the elliptic type, given by Eq. (2.11) § ∂ 2 u ∂ 2u u u· + 2 = H 4 ¨ ξ ,η , u , , ¸ 2  ∂ η ∂ξ ξ ∂η ¹ ©

(2.11)

Case 3: Parabolic equation (B2 – AC C) = 0 For this case, it can be seen from Eq. (2.21) that only one family of real characteristics exists. Because of this fact, we can set in Eq. (2.17) for example η = x. Note that this is only possible if ξ depends on y, so that the Jacobian, defined by Eq. (2.13), is not zero. Then we obtain immediately from Eq. (2.17) that C = 0 while B is equal to:

B

A

∂ξ ∂x

B

ξ

(2.23)

∂y

This expression is identical to zero, as it can be deduced from Eq. (2.18) rewritten as follows - for the case (B2 – AC C) = 0 § ∂ξ · A¨ ¸ © ∂x

2

§ ξ ·§ ξ · 2 ¨ ¸¨ ¸ x © y¹

B2 § ξ · ¨ ¸ A © y¹

2

1§ ¨ A©

ξ x

ξ· ¸ y¹

2

(2.24) 0

2.1 Classification of Second-Order Partial Differential Equations

17

Finally, the standard form of the parabolic equation, given by Eq. (2.10), is obtained if we divide Eq. (2.16) by A . In order to illustrate the above shown classification, we will investigate some simple examples: Example 1: The equation

∂ 2u ∂ 2u + 2 ∂x∂y ∂x 2

∂ 2u ∂y 2

(2.25)

0

is parabolic, because the expression B 2 tion (2.21) reduces to

AC = 1 1 0 . The characteristic equa-

dy =1 dx

(2.26)

y x C1

(2.27)

which has the solution

If we make now a change of coordinates according to

ξ=y x η=x

(2.28)

where η has been selected arbitrarily, although always respecting the condition that the Jacobian, defined in Eq. (2.13), is not equal to zero. Introducing the new coordinates into Eq. (2.25), we obtain ∂ 2u =0 ∂η 2

(2.29)

This equation has the general solution u

F(

)

( )

(

)

(

2 yx 2

∂u ∂x

0

)

(2.30)

Example 2: Consider the equation

y4

∂ 2u ∂x 2

x4

∂ 2u ∂y 2

(2.31)

From this equation, we obtain B 2 AC x 4 y 4 > 0 . This shows that Eq. (2.31) is hyperbolic everywhere except along the coordinate axis ((x = 0 orr y = 0). From Eq. (2.21) we obtain the following two ordinary differential equations for the characteristics

18

2 Linear Partial Differential Equations

dy = dx

(

)

x2 y 2 § x · A= 4 =¨ ¸ y © y¹

2

(2.32)

and

§x· dy = −¨ ¸ dx © y¹

2

(2.33)

From the two Eqs. (2.32 - 2.33) we calculate the characteristic curves to be: 1 ( 3

)

1 ( 3

C1 ;

)

C2

(2.34)

In order to transform Eq. (2.31) into its standard form, we have to perform the following coordinate transformation

1 ( 3 1 η= ( 3

(2.35)

)

ξ=

)

Using Eqs. (2.14-2.15), one obtains

−4 x 4 y 4 § −2 y 2 ¨ ©

∂ 2u − 2 ( xy + yx ∂ξ∂η 2

u ∂ξ

2

u· ¸

η¹

) ∂∂ξu + 2 (

) ∂∂ηu

(2.36)

0

and after simplifying and replacing x,y by ξ and η we finally get

∂ 2 u 1 1 ∂u 1 3ξ − η ∂u − − =0 ∂ξ η 3 ξ η ξ 3 η 2 ξ 2 ∂η

(2.37)

Example 3: We consider the wave equation

∂2Φ 2 ∂2Φ −c =0 ∂t 2 ∂x 2

(2.38)

This equation describes for example the one-dimensional propagation of sound in a pipe. Because B 2 AC > 0 , the equation is hyperbolic in the region of interest. The characteristic equations are given by

(

dt = B + B 2 − 4 AC dx

) 2 A = −22cc

2

=−

1 c

(2.39)

2.2 Character of the Solutions for the Partial Differential Equations

19

and (2.40)

dt 1 = dx c

From these two equations we obtain ct following new coordinates:

C1 and −ct

x

ξ = x ct η = x ct

x

C2 and we get the

(2.41)

If we introduce this new coordinates into Eq. (2.38), we get the following simple partial differential equation

∂2Φ =0 ∂ξ η

(2.42)

which has the general solution

Φ(

)

1

( )

2

(ξ )

(2.43)

or rewritten in x,tt coordinates Φ ( , ) = Ψ1 (

)

(

)

(2.44)

This shows that the solution of Eq. (2.38) can be expressed as the superposition of two waves, which move with constant velocity c into different directions of the solution domain. This shows also nicely how the information in the problem is transferred by the two real characteristics. The solution obtained here is known in literature as the d´Alembert solution.

2.2 Character of the Solutions for the Partial Differential Equations In the preceding section we have concentrated on the classification of the different second-order partial differential equations. However, for solving actual physical problems, it is of great importance to discuss also the character of the solutions and the associated boundary conditions. 2.2.1 Parabolic Second-Order Equations

Let us start our discussion with the parabolic partial differential equation. As stated before, this equation has one real characteristic. As an example, we look at the heat conduction equation for a one-dimensional unsteady conduction problem

20

2 Linear Partial Differential Equations

in a slab (see Fig. 2.1). The slab has the thickness l and the spatial coordinate ranges from 0 < x < l.

Fig. 2.1: Transient heat conduction in a slab

Assuming that the material properties of the slab are constant, the temperature distribution in the slab can be obtained from the solution of the following equation

ρc

∂T ∂t

k

∂ 2T ∂x 2

(2.45)

The temperature distribution of the slab at the beginning of the process (t = 0) is given. This is the initial condition of the problem T(

)

( )

(2.46)

In addition to this initial condition, boundary conditions have to be prescribed at the surface of the slab for x = 0 and x = l. Here the following different types of boundary conditions are possible: • Boundary conditions of the first kind (Dirichlet wall boundary conditions). Here the temperature at the boundary is specified, for example T(

)

()

( )

()

(2.47)

• Boundary conditions of the second kind (Neumann conditions). For this type of boundary conditions the gradient is specified at the boundaries, for example § ∂T · ¨ ¸ © ∂x ¹ x

= f4 ( 0

)

§ ∂T · ¨ ¸ = f5 ( x ¹ x =l

)

(2.48)

• Boundary conditions of the third kind. Here a combination of temperature and gradient is prescribed at the surface. Such boundary conditions are relatively common in technical systems, since they describe, for example, the case of a

2.2 Character of the Solutions for the Partial Differential Equations

21

slab which is heated or cooled by a fluid at temperature T1 or T2 flowing over the boundaries of the slab. A typical example is § k ¨¨ T © x

· ¸¸ ¹ x=0

h1 § T ( t

§ k ¨¨ T © x

· ¸¸ ¹x l

h 2 § T ( t l ) T 2 ·¸

©

)

T1 ¸· ,

©

(2.49)

¹

¹

where h1 and h2 are heat transfer coefficients. Of course, all the above mentioned boundary conditions can be present in any possible combination, for example: at x = 0, a constant wall temperature is prescribed; whereas, at x = l, a boundary condition of the third kind is applied. Summarising the above discussion, it can be seen that for the parabolic secondorder partial differential equation an initial condition together with boundary conditions at the surface need to be specified. This means that the temperature at an arbitrary point P in the domain (see Fig. 2.1) is always influenced by the wall boundary conditions at x = 0 and x = l. In addition, all disturbances, which are specified for t = 0, will propagate into the solution domain for all subsequent times. On the other hand disturbances, which are introduced at a later time t1, can not influence the solution for t < t1. This shows nicely the character of the solution which depends only on one real characteristic. 2.2.2 Elliptic Second-Order Equations

For the elliptic equation, the characteristic curves are families of conjugate complex functions. In order to investigate the character of the solutions and the boundary conditions needed for this type of equations, we select as an example the steady-state heat conduction in a square plate ( ) , as shown in Fig. 2.2.

Fig. 2.2: Steady-state heat conduction in a square plate

22

2 Linear Partial Differential Equations

The steady-state temperature distribution is obtained from the solution of the following equation: ∂ 2T ∂ 2 T + =0 ∂x 2 ∂y 2

(2.50)

For this type of equation, boundary conditions have to be prescribed along each point of the boundary. We might illustrate this for the above given example. Here the following different types of boundary conditions can be assigned: • Boundary conditions of the first kind (Dirichlet conditions). The temperature is specified at each point of the boundary T(

) )

T(

( ) ( )

( (

) )

( ) ( )

(2.51)

• Boundary conditions of the second kind (Neumann conditions). The heat flux normal to the wall is specified along the boundary. § ∂T · ¨ ¸ © ∂xx ¹ x § T· ¨ ¸ © ∂yy ¹ y

= f1 (

)

§ ∂T · ¨ ¸ = f2 ( x ¹x=a

)

)

§ T· = f4 ( ¨ ¸ ∂yy ¹ y = a

)

0

= f3 ( 0

(2.52)

• Boundary conditions of the third kind. This might be again a combination of the normal gradient at the surface and the temperature. One example is:

§ ∂T · ¸ © ∂∂xx ¹

f2 (

x 0

§ ∂T · ¸ ∂∂xx ¹

f4 (

y 0

∂T ) T ) , k § ¸· ∂∂yy ¹

f1 (T ( y ) T1 ) , k

k¨

§ ∂T · k¨ ¸ © ∂∂yy

(

)

2

)

x=a

f3 ( T ( x

3

(2.53)

(

)

4

)

y=a

Again, the boundary conditions can be applied as mixed boundary conditions. From the above examples, it can be seen that for an elliptic equation we have to deal with a boundary-value problem, whereas for the parabolic equation, we had to solve a combined initial, boundary-value problem. This means that for the elliptic problem any disturbance, which is brought into the region of interest (for example by slightly changing one boundary condition) will immediately influence the solution of the problem at a given point in the domain (see point P in Fig. 2.2).

2.2 Character of the Solutions for the Partial Differential Equations

23

2.2.3 Hyperbolic Second-Order Equations

Hyperbolic partial differential equations mainly appear in vibration and wave problems. These equations have two real characteristics. The one-dimensional wave equation for a perfectly flexible string serves here as an example to explain the character of the solution and the associated boundary conditions (see Fig. 2.3).

Fig. 2.3: Flexible string

The differential equation is given by ∂ 2u 1 ∂ 2u − =0 ∂x 2 b 2 ∂t 2

(2.54)

where b is a constant. As shown in the previous section (see Eq. (2.44)), the general solution of this equation is given by u(

)

(

)

(

)

(2.55)

Therefore, it would be easiest to prescribe boundary conditions for Eq. (2.54) along two parts of the characteristics intersecting at one point. This would be a complete initial value problem. However, for most physical problems, this description of the boundary conditions is not typical. Instead, the following boundary conditions might be normally applied: • Boundary conditions of the first kind. Here the deflection of the string at the location x = 0 and x = l might be prescribed. u(

)

()

( )

()

(2.56)

If the string is fixed at the locations x = 0 and x = l, f1 and f2 will be zero. • Boundary conditions of the second kind. Here we will prescribe ∂u/∂∂x for x =0 and x = l. • For the boundary conditions of the third kind, we will specify a combination of u and ∂u/∂∂x for x = 0 and x = l. As for all other types of equations, the boundary conditions can also be applied in a mixed form.

24

2 Linear Partial Differential Equations

In addition to the two boundary conditions at x = 0 and x = l, initial conditions for t = 0 have to be specified for the problem. These initial conditions for the finite string could be that u ( ) ( ) ( ) ( ) .

2.3 Separation of Variables This section is devoted to an introduction into the method of separation of variables. This method is one of the most commonly used methods for solving linear partial differential equations. The method is explained in the following sections by two basic examples. In the next chapters, more advanced problems are considered. 2.3.1 One-Dimensional Transient Heat Conduction in a Flat Plate

We consider the energy equation for heat conduction in a flat plate. The problem is depicted in Fig. 2.4. The plate has thickness δ and length L. At the two surfaces x = 0 and x = δ the plate is subjected to constant temperatures.

Fig. 2.4: Transient heat conduction in a flat plate

Under the assumption that the material properties of the plate are constant, the energy equation takes the following form.

ρc

∂T ∂t

2 § 2T T· k¨ 2 + 2 ¸ x y © ¹

(2.57)

We now assume that the dimension L is much larger than δ, so that the heat conduction in the y-direction is negligible compared to the heat conduction in the xdirection. Therefore, the problem simplifies to ∂T ∂ 2T =a 2 ∂t ∂x

(2.58)

where /( ρ ) is the thermal diffusivity of the material. Eq. (2.57) has to be solved together with the following boundary conditions

2.3 Separation of Variables

x

0 :T

T1

x

δ :T

T2

25

(2.59)

and the initial condition t

0 :T

T0 (

)

(2.60)

This problem is described by a parabolic equation. This means that one real characteristic exists for the solution. Before solving the above given problem, we first introduce the dimensionless quantities x =

(2.61)

T T1 at , t = 2 , Θ = T2 T1 δ δ x

This results in the following problem ∂Θ ∂ 2 Θ = ∂t ∂x 2

(2.62)

with the boundary conditions x = 0 : Θ = 0 x = 1: Θ = 1 t

0: Θ

(T ( x )

(2.63) T )/(

)

( )

Before applying the method of separation of variables to the Eqs. (2.62 – 2.63), we want to investigate the solution domain, shown in Fig. 2.5. Here the parabolic nature of the problem is clearly visible. The initial condition for  0 is propagated into the solution domain for larger times. Any disturbance introduced into the problem at t t1 will therefore only influence the solution at subsequent times. The solution for t t1 stays unchanged.

Fig. 2.5: Solution domain for the transient conduction in the plate

Let us assume that the solution of the problem can be expressed in the form

26

2 Linear Partial Differential Equations

Θ = H ( )G (

)

(2.64)

Introducing Eq. (2.64) into the boundary conditions, results in: x

0 : H (t ) G ( 0)

x 1: H (

(2.65)

0

) (1)

1

From this equation, we notice immediately that using the expression (2.64) can not result in a solution to the present problem, because the boundary condition for x = 1 can not be satisfied if H ( ) is an arbitrary function of t . Therefore, we conclude that we first have to make the boundary conditions homogenous, in order to find a solution with the help of Eq. (2.64). This can be done by splitting the solution into two parts

) + ΘT (

Θ = ΘS (

)

(2.66)

The steady-state solution ΘS is simply a linear distribution and is given by ΘS (

) = x

(2.67)

Introducing Eq. (2.66) into the Eqs. (2.62-2.63) results in the following problem for ΘT ∂ΘT ∂ 2 ΘT = ∂t ∂x 2

(2.68)

with the boundary conditions x = 0 : ΘT = 0 x = 1: ΘT = 0 t = 0 : ΘT = Θ0 ( x ) − Θ

(2.69)

(

=

) = Θ0 ( ) − x

Introducing now again the product of functions, given by Eq. (2.64), we obtain the boundary conditions x x

0: H (

) (0) 1: H ( ) (1)

0

=>

0

=>

G(

) G( )

0

(2.70)

0

This shows that the expression given by Eq. (2.64) is able to satisfy the two boundary conditions of the problem. Therefore, this approach promises to be successful. Introducing Eq. (2.64) into the partial differential equation (2.68), one obtains H '(  ) ( )

''(( ) (  )

By separating the variables, this equation can be rewritten as

(2.71)

2.3 Separation of Variables

H ′( ) = H ( )

′′( ) ( )

27

(2.72)

~

The left hand side of this equation is only a function of t , whereas the right hand x . Therefore, both sides of the equation must be conside is only a function of ~ stant. This constant is set to C1. H ′( ) = H ( )

′′( ) = C1 ( )

(2.73)

Let us first investigate the differential equation for the function H H. This equation takes the form H ′(  ) = C1 H ( )

(2.74)

and can easily been integrated to give H ( )

2

exp (

)

(2.75)

If we have a closer look at this equation, it can be seen that the function H (  ) tends to infinity for t → ∞ if C1 > 0 . However, this would not lead to a physically meaningful solution for the problem, because the temperature would tend to infinity for large times. For C1 = 0 , Eq. (2.75) results in a constant for H (  ) and the time dependence of the solution would be lost. Therefore, we can conclude that the constant C1 must always be smaller than zero for the problem under consideration. This can be expressed by replacing the constant by C1 = −λ 2 . Then we obtain for the function H H ( ) For the function from Eq. (2.73)

2

exp (

)

(2.76)

( ) , one obtains the following ordinary differential equation G ′′( ) = −λ 2 G(( )

Ÿ

G ′′ λ 2 G = 0

(2.77)

This equation has to be solved together with the homogeneous boundary conditions given by Eq. (2.70). It has the trivial solution G = 0 and will have further solutions for selected values of λ. These selected values of λ are called the eigenvalues of Eq. (2.77). The problem given by Eq. (2.77) and associated boundary conditions, see Eq. (2.70), is called an eigenvalue problem. This sort of problem is discussed in more details in Chap. 3. The general solution of Eq. (2.77) is given by G(( )

3

sin( i (λ )

4

cos((λ )

(2.78)

28

2 Linear Partial Differential Equations

where the solution must satisfy the two boundary conditions G(0) (0) 0, 0 From the boundary condition

(0)

(2.79)

(1) 0

0 , it follows that C4 is zero and one obtains

C3 sin(λ )

G

(2.80)

Now, if we apply the second boundary condition 0

(1)

0 , we find

sin(λ 1)

3

(2.81)

Eq. (2.81) shows that either C3 has to be zero (which would be the trivial solution of the problem, where G = 0) or that sin(λ) has to be zero. The latter is only possible if

λ

nπ

with

n=1,2,3,...

(2.82)

These special values of λ are the eigenvalues of Eq. (2.77) and are shown in Fig. 2.6.

Fig. 2.6: Eigenvalues for the eigenfunction sin(λ)

The solution for ΘT is obtained from the Eqs. (2.75) and (2.80) as ΘT = C2 exp (

)

3

sin( si (λ )

(2.83)

For simplicity, we combine the constants C2 and C3 and have ΘT = C sin(λ ) exp e p(

)

(2.84)

Now we try to fulfill the initial condition using Eq. (2.84). Inserting Eq. (2.84) into Eq. (2.69) results in

t = 0 : ΘT = Θ 0 ( => Θ0 (

)

)

( ) expp ( ( )

)

(2.85)

From the equation above, one can see that Eq. (2.84) is automatically a solution of the problem if

2.3 Separation of Variables

Θ0 (

)

(

),

1, 2,3,...

29

(2.86)

However, from Eq. (2.82) it is clear that there is an infinite number of eigenvalues. Because the partial differential equation is linear, we use the principle of superposition to construct the final solution of the problem. This means that the solution will be given by ∞

ΘT = ¦ Cn sin( si (λn ) exp e p(

)

n =1

(2.87)

This solution has to fulfil the initial condition. Inserting Eq. (2.87) into Eq. (2.69) results in ∞

¦

Θ0 ( )

n

(2.88)

si (λn ) sin(

n =1

which means that we have to represent the function Θ0 ( ) − x by a Fourier series (see Stephenson (1986), Myint-U and Debnath (1987), Zauderer (1989) or Sommerfeld (1978)). In order to obtain the unknown coefficients Cn, from Eq. (2.88), we multiply both sides of the equation by sin(λm ) and integrate the resulting expressions across the region of interest for x between zero and one. This results in 1

³(

i (λm ) sin(

1 ∞

³¦

)

n

sin((λn )sin( ))si (λm )d dx

(2.89)

0 n =1

0

Exchanging the summation and integration signs on the right hand side of this equation leads to 1

³(

si (λm ) sin(

∞

1

s (λ ¦ ³ sin(

)

n

1

0

m

) sin( si (λn )d dx

(2.90)

0

If we now evaluate the integrals on the right side of Eq. (2.90), we find that 1

³ sin(λ

m

= 0 for n ≠ m

)sin( ))si (λn ) ddx

(2.91)

0

1

= ³ sin 2 (λn )ddx = 0

1 2

for

n

m

Writing Eq. (2.90) in detail gives 1

³( 0

i ( ) sin(

1

) ddx

= C1 ³ sin(m x ))sin( )si ( x)ddx 0

(2.90)

30

2 Linear Partial Differential Equations

1

+C2 ³ sin(m x))sin(2 )si (2 ( x)dx d 0

+ 1

+Cn ³ sin 2 (n x)dx d (forr n = m) 0

+ 1

+Cα ³ sin(m x))sin( )si (

x) dx d

0

+…

From Eq. (2.91) one can see that in the sum on the right hand side of this equation only the term containing Cn will be non zero. Therefore, Eq. (2.90) reduces to 1

³(

1

si ( ) sin(

)

n

0

si ³ sin

2

(

(2.92)

)d dx

0

From Eq. (2.92), the unknown constants Cn can be evaluated to be 1

Cn

³(

si ( ) sin(

) ddx

0

1

2 ³ sin (

) ddx

(2.93) 1

2³(

( )

i ( ) sin(

)d dx

0

0

The final solution of the problem is given by Eq. (2.66) with the steady-state solution according to Eq. (2.67) and the transient solution according to Eq. (2.87). Thus, ∞

Θ = x + ¦ Cn sin( i ( n =1

) exp p(

)

(2.94)

In order to show the transient evolution of the temperature field, we select Θ0 ( ) 1 for the above example. Using this initial condition, Eq. (2.93) becomes 1

Cn

2 ³ (1 x ) sin( si (n x)dx d = 0

2 nπ

and the temperature distribution in the solid is given by

(2.95)

2.3 Separation of Variables

∞

) exp e p(

2 sin( i ( n =1 nπ

Θ = x + ¦

31

(2.96)

)

This temperature distribution is shown in Fig. 2.7 for different times. One can see nicely that the temperature distribution in the solid changes from the prescribed constant initial temperature distribution to the linear temperature shape for the steady-state temperature distribution for t → ∞ . 1.1

Θ

1

0.9

∼ t = 0.005

0.8 0.7

∼ t = 0.01

0.6 0.5

∼ t = 0.1

0.4

∼ t=1

0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 ∼ 1

x

Fig. 2.7: Transient temperature distribution in the plane wall for selected times

In addition, one can see from Eq. (2.96) that the individual parts of the sum in this equation are decaying rapidly with increasing time (notice that the argument of the exponential function contains (nπ )2 as a multiplier). 2.3.2 Steady-State Heat Conduction in a Rectangular Plate

As a second example to explain the method of separation of variables, we investigate the heat conduction in a rectangular plate, which has height c and width b (see Fig. 2.8). All four sides of the plate are set to a constant temperature T0. Inside the rectangular area, a sink is located with constant sink intensity K. Assuming constant physical properties, the energy equation for this steady-state heat conduction problem is given by § 2T 0 = k¨ 2 © ∂x

T· ¸ K ∂y 2 ¹ 2

(2.97)

32

2 Linear Partial Differential Equations

Fig. 2.8: Geometrical configuration and boundary conditions for the heat conduction problem in a flat plate

with boundary conditions T (b

T (x c

y) T

)

T

T( b

y ) T0

T (x c

)

(2.98)

T0

As for the first example, before proceeding with the solution of the problem, we first introduce dimensionless quantities. Suitable dimensionless quantities are given by Θ=

T T0 x , x = , T0 c/2

y=

y /2

(2.99)

Introducing these quantities into Eqs. (2.97-2.98) results in 0= Θ( Θ(

) )=

∂2Θ ∂2Θ + +K ∂x 2 ∂y 2

( Θ(

(2.100)

)

−

0

(2.101)

)=0

where the following abbreviations have been used: K

K c2 b , A= 4kT0 c

(2.102)

This problem is described by an elliptic second-order partial differential equation. The boundary conditions, expressed by Eq. (2.101), are homogeneous, but the differential equation (2.100) is not.

2.3 Separation of Variables

33

In order to find a solution of the problem, we make once again use of the method of superposition, since the partial differential equation is linear, and split the solution into two parts Θ = Θh + Θ p

(2.103)

Θ h represents the solution of the homogeneous differential equation (without a sink, K = 0) and Θ p is one particular solution of the problem. Let us first focus on the special solution of the problem. In order to find this solution, we assume that Θ p = f ( y ) . Alternatively, we could also assume p ( ) and obtain the same final solution of the problem. The analysis, however, would be altered. If we substitute p f ( ) , into Eq. (2.100), the following relation for the function f ( y ) is obtained f ′′( y )

K

1 K y2 2

f ( y)

(2.104)

C1 y C2

Since we need only one special solution of the problem, we could set C1 and C2 equal to zero. However, a better choice is to select the constants C1 and C2 in such a way that the two boundary conditions p ( ,1) ,1)) 0, 0, 0 are satisfied. p ( , 1) If we do so, we obtain the following solution of the problem Θp =

(2.105)

K ª1 y 2 º¼ 2¬

After having obtained the solution for Θ p , one has to solve the following problem for Θ h which is deduced from the Eqs. (2.100 –2.101) 0= Θh ( Θh (

) )

/ 2 (1 0,, 0

∂ 2 Θ h ∂ 2 Θh + ∂x 2 ∂y 2

),

( Θh ( h

(2.106)

, y) −

)=0

/ 2(

)

(2.107)

Note that the differential equation for Θ h is now homogeneous, whereas two of the boundary conditions are non-homogeneous. Furthermore, note that the two boundary conditions, corresponding to a constant value of y , are still homogeneous. This is of importance for the subsequent analysis of the problem. We now assume that Eq. (2.106) has a solution, which can be obtained by the method of separation of variables. Thus Θh = F ( ) ( ) Introducing Eq. (2.108) into the Eqs. (2.106-2.107) results in

(2.108)

34

2 Linear Partial Differential Equations

F ''( x) G ( y ) G ''( y ) F ( x)

0

(2.109)

from which we obtain (2.110)

F ''( x) ''(( ) =− = ±λ 2 F ( x) G( y)

From Eq. (2.110), one notices that a physically plausible solution occurs for both + λ 2 and −λ 2 . In order to investigate this problem further, we analyse in more details the solutions for the function ( ) . From Eq. (2.110) we get G1 ''( ) λ 2

1

( )

0

for + λ 2

(2.111)

G2 ''( ) λ 2

2

( )

0

for −λ 2

(2.112)

which gives rise to the following two possible solutions G1 ( )

cos((λ )

4

sin( i (λ )

(2.113)

cosh( h(λ )

4

sinh( i h(λ )

(2.114)

3

G2 ( )

3

If we now reconsider the problem to be solved (Eqs. (2.106-2.107)), one can see that h ( ) ( ) has to be zero for y = ±1 . If we satisfy these two boundary conditions by Eq. (2.114), we obtain only the trivial solution, because the functions cosh(λ ) andd sinh( i h(λ ) have only one zero point. Instead, if we satisfy the two boundary conditions by Eq. (2.113), we obtain an equation, which determines the eigenvalues. Therefore, Eq. (2.113) is the desired solution and thus, we have to select + λ 2 in Eq. (2.110). For the function ( ) , one obtains from Eq. (2.110) F ′′( ) λ 2 ( )

0

(2.115)

sinh( i h((λ )

(2.116)

which has the solution F( )

5

cosh( h(λ )

6

Combining the solutions for F and G leads to the following expression for Θ h Θh = (

) (

)

(2.117)

This expression has to satisfy the boundary conditions given by Eq. (2.107) Θh ( Θh (

) )

/ 2 (1 00,,

),

( Θh ( h

, y) −

)=0

/ 2(

)

(2.107)

2.3 Separation of Variables

35

Applying the two boundary conditions for fixed values of y , the following two equations are obtained C3 cos (

) C3 cos ( ( 1) )

( ) si ( ( 1) ) 4 sin

Because cos(λ ) cos( λ ) and sin(λ ) equations that C4 = 0 and that C3 cos (

)

(2.118)

0 0

sin( λ ) , one obtains from the above (2.119)

0

From this equation it follows that

λ

2n 1 π, 2

(2.120)

n 1, 2 ,3,...

and the following solution for Θ h is obtained

(

Θ h = C3 cos((λ )

)

(2.121)

From the two boundary conditions, corresponding to a fixed value of x in Eq. (2.107), it can be seen that h ( , ) , ) , which indicates Θ h is an even h( function in x . Therefore, it follows that C6 = 0. Thus Θ h = C cos((λ y ) cosh( h(λ x ) ,

C

C3C5

(2.122)

Since an infinite number of eigenvalues has been found from Eq. (2.120), the solution for Θ h can be constructed by superimposing all these individual solutions. This results in ∞

Θ h = ¦ Cn cos(λn ) cosh( h(λn )

(2.123)

n =1

The unknown coefficients Cn can be obtained by matching the boundary condition Θh ( , ) / 2 (1 2 ) by Eq. (2.122). This results in −

∞

K (1 y 2 ) 2

¦

n

cos((λn y ) cosh( h(λn )

(2.124)

n =1

If we multiply both sides of the above equation by cos(λm ) and integrate the resulting expressions between –1 and 1 we obtain: −

K 2

1

³ (1

−11

1

2

) cos((λm ))dy dy d

§

³ ¨© 1

∞

· Cn cos((λn y ) cos((λm y ) cosh( h((λn A) ¸ d dy n=1 ¹

(2.125)

36

2 Linear Partial Differential Equations

Again, the summation and integration signs on the right hand side of Eq. (2.125) can be interchanged. Then, it is obvious that from the sum only one term will not be equal to zero, because the integral 1

³ cos(λ

) cos((λm )

n

for λn

0

λm

(2.126)

−1

for λn

= 1

λm

Finally, the following equation is obtained for the determination of the unknown coefficients

− Cn =

K 2

(2.127)

1

³ (1

−1

y 2 ) cos((λn y )ddy =

cosh(λn )

2 ( 1)

n

λn3 cosh(λn )

The solution of the problem, given by Eqs. (2.100)-(2.101), is obtained by combining the two parts of the solution Θ h and Θ p . This gives finally Θ=

K§ 2 ¨1 − y 2©

· 4 ( 1)) n cos((λn y ) cosh( h((λn x) ¸ 3 h( n ) n =1 λn cosh( ¹ ∞

(2.128)

The solution obtained here shall serve as an example that a lot of solutions, which are obtained for heat conduction problems, can be very useful for other applications. To that aim, let us investigate the flow in a rectangular channel. The channel has width b and height c. The geometry under consideration is shown in Fig. 2.9. For this problem, u, v, w are the flow velocities in the x,y and z direction. Under the assumption of a steady, incompressible, laminar flow with constant fluid properties, the Navier-Stokes equations reduce to §

§

2

2 2 u u u· + 2 + 2¸ 2 ∂ x ∂ y ∂ z © ¹

(1.1)

§

2

(1.2)

v

u ∂y

w

u· ¸ ∂z ¹

Fx

∂pp ∂x

μ¨

v

v ∂y

w

v· ¸ ∂z ¹

Fy

∂pp ∂y

μ¨

w w w· v w ¸ ∂y ∂z ¹ © ∂x

Fz

∂pp ∂z

μ¨

u ∂ © x

ρ ¨u §

v © ∂x

ρ ¨u §

ρ ¨u

2 2 v v v· + + ¸ 2 2 ∂y ∂z 2 ¹ © ∂x

§

2 2 w w w· + + ¸ 2 2 ∂y ∂z 2 ¹ © ∂x 2

(1.3)

and the mass continuity equation is ∂u ∂v ∂w + + =0 ∂x ∂y ∂z

(1.6)

2.3 Separation of Variables

37

Fig. 2.9: Geometrical configuration and coordinate system for the flow in a rectangular channel

If we now further assume that the flow is hydrodynamically fully developed, i.e. the velocity profile does not change along the axial direction, only the w component of the flow velocity is present. The u and v components are identically zero. Additionally, the w component of the flow can only be a function of the x and y coordinate for the hydrodynamically fully developed flow. Then the problem simplifies to (see for example Spurk (1987)) 0=−

2 § 2w ∂pp w· +μ¨ 2 + 2 ¸ ∂zz ∂y ¹ © ∂x

(2.129)

where the pressure gradient in the axial direction is constant for a hydrodynamically fully developed flow. The boundary conditions are the no slip conditions at all boundaries of the channel. Thus w(b / 2, y ) 0, 0

(

/ 2, 2 ) 0

w(( , / 2) 0, 0

( ,

/ 2) 0

(2.130)

If we introduce into Eq. (2.129) the abbreviation K = −1 / μ ∂p / ∂z , it can be seen that the problem for determining the fully developed velocity profile is identical to the heat conduction problem in a plate containing a heat sink with constant sink intensity (the derivation of the fully developed velocity field in a rectangular channel is given for example in Spurk (1987)). This shows nicely the similarity of the equations describing problems in Fluid Mechanics and Heat Transfer. 2.3.3 Separation of Variables for the General Case of a Linear Second-Order Partial Differential Equation

At the beginning of this chapter, we have been concerned with a linear second order partial differential equation which depended on the two variables x and y. The most general form of this homogeneous equation is given by

38

2 Linear Partial Differential Equations

∂ 2u ∂ 2u 2 , ( ) ∂x 2 ∂x∂y ∂u ∂u +D D ( x y) ( x, y ) ∂x ∂y A( x y)

(

)

∂ 2u + ∂y 2

( x, y ) u

(2.1)

0

where A,...,F F are constants or functions of x and y which are sufficiently differentiable in the domain of interest. In the previous two examples, we used the method of separation of variables to derive a solution of the linear partial differential equation as an infinite sum. However, we only addressed very special cases of Eq. (2.1). It is now interesting to evaluate, under which conditions a separation of variables is possible for Eq. (2.1). In order to answer this question, we consider the transformed equation (2.16) ∂2u ∂ξ 2 ∂u ) ∂ξ

A(

)

D(

(

2

(ξ , η )

∂ 2u

)

,

ξ η

∂u

η

(

∂ 2u

(

)

)

0

η

2

(2.16)

+

where the new coordinates ξ, η, defined by Eq. (2.6), have been used. Let us substitute H(

u

) ( )

(2.131)

into Eq. (2.16). From this we obtain A(

D(

) H ( )G( ) ) ( ) ( )

2B (

) H ( )G ( ) ) ( ) ( )

(

C(

) ( ) ( )+ ( ) ( ) 0

(2.132)

where the prime indicates the differentiation of the functions

(ξ ) and ( ) with

respect to the independent variable. Dividing Eq. (2.132) by H (

) ( ) results in

A( D(

) H( ) H ′( ) ) H( )

)

H ′′ (

2

(

,

(ξ ,η )

)

H ′ (ξ ) G ′ (η )

(

H (ξ ) G (η )

G′ ( G(

) )

(

)

)

G ′′ (

) + G( )

(2.133)

0

From the above equation it can be seen that the variables can only be separated if B( ) 0 . According to Eq. (2.17), this requires that the new coordinates are chosen in a way to ensure that B=A

§ ξ η ∂ξ η ξ η· ξ η + B¨ + =0 ¸+C ∂x ∂x ∂ x ∂ y ∂ y ∂ x ∂ y ∂y © ¹

(2.134)

2.3 Separation of Variables

After setting B (

)

39

0 , Eq. (2.132) can be rewritten in the following way

A(

) H ′′ (ξ ) D (ξ ,η ) H ′ (ξ ) C (ξ η ) G ′′ ( ) + + + N( ) H ( ξ ) N (ξ , η ) H ( ξ ) N ( ξ η ) G ( ) E( ) G ′ (η ) F ( ) + + =0 N( ) G (η ) N ( )

(2.135)

where the whole equation has been divided by the function N ( assume that

) . If we further

F(

N(

) )

= f1 (

)

(2.136)

( )

Eq. (2.134) can be written as A(

) H ′′ (ξ ) D (ξ η ) H ′ ( ) + + f ( )= N( ) H ( ξ ) N (ξ η ) H ( ) 1 C( ) G′′ (η ) E (ξ η ) G′ ( ) − − − f ( ) = const. N( ) G (η ) N (ξ η ) G ( ) 2

(2.137)

The left side of this equation should now only be a function of ξ and the right side of the equation should only be a function of η. This is only possible, if the following restrictions are satisfied: A(

N( C(

N(

) = f ( ) ) 3 ) = f ( ) ) 5

D(

N( E(

N(

) ) ) )

f4 (

),

f6 (

)

(2.138)

The present analysis might be very helpful in order to check in advance if the method of separation of variables can be applied to the problem under consideration. It would be incorrect, however, to assume that the method leads to a solution in all cases, where the separation of variables is possible. Problems

2.1

Consider the partial differential equation 4

∂ 2u ∂ 2 u ∂ 2 u ∂u ∂u 5 + 2+ + =2 ∂x∂y ∂x ∂y ∂x 2 y

a.) Determine the type of the differential equation.

40

2 Linear Partial Differential Equations

b.) What are the characteristics of this equation? c.) Transform the equation into its standard form. d.) Determine the general solution of the above given differential equation (hint: use the substitution: = ∂ / ∂η ). 2.2

Consider the partial differential equation −x

∂ 2u ∂x 2

y

∂ 2u ∂y 2

1 ∂u 2 ∂x

A(

)

∂u ∂y

0

Solve this equation using the method of separation of variables. F(x)G(x ( ) into the equation. How should the function a.) Insert u = F( A(x, ( y) look like, so that the method of separation of variables can lead to a solution of the above equation? b.) Determine the type of the partial differential equation. Show the result in a x-y - diagram for −∞ < x < +∞ and −∞ < y < +∞ . c.) Calculate for x > 0, y > 0 the characteristics of the equation and transform the equation into its standard form (use for this A(x, y) = -1/2). 2-1.

Consider the partial differential equation

∂ 2u ∂ 2u +5 2 ∂x∂y ∂x

4

∂ 2u ∂u 10 2 ∂yy ∂y

sin x

a.) Determine the type of the differential equation. b.) What are the characteristics of this equation? c.) Transform the equation into its standard form. 2-2.

A thin rectangular plate with sides of length a and b is subjected to a constant temperature T1 at three sides ( ), whereas the remaining side of this plate is subjected to the temperature distribution ­ 3 2π y ½ T (a, y ) T1 ®sin i ¨ ¸ 1¾ © b ¹ ¿ ¯

We are interested in the steady-state temperature distribution in the plate. All material properties of the plate are constant. The steady-state temperature distribution can be calculated from the energy equation

∂ 2T ∂ 2 T + =0 ∂x 2 ∂y 2 with the above given boundary conditions.

2.3 Separation of Variables

41

a.) Make the energy equation and the boundary conditions dimensionless. b.) Solve the problem using the method of separation of variables. 2-3.

Consider a slab, which has extensions in the y- and z-direction much bigger than in the x- direction. The slab has constant initial temperature Ti. At t = 0, the slab is exposed at both sides ( x 00, x = δ ) to convective cooling. The temperatures of the surrounding fluid are given by TC and TG. The problem under consideration is described by the following, simplified energy equation

ρc

∂T ∂t

k

∂ 2T ∂x 2

and the following boundary conditions t

0 :T

x

0 : hG (TG T ( x

x

δ:

Ti

G

(

C

(

0)) k

δ )))

∂T ∂xx ∂T ∂xx

=0 x=0

0 x =δ

a.) Introduce dimensionless quantities into the above equations. Use Θ= Bi G

TG T at k x , t = 2 , a = , x= , TG Ti δ ρc δ hGδ hδ , Bi C = C k k

b.) Split the solution of the problem into the steady-state solution and into the solution of the transient part. Show that for the transient part of the solution, the two boundary conditions for 0 and 1 are homogeneous. c.) Solve the first problem. What is the steady-state temperature distribution in the slab? d.) Solve the transient problem. What is the complete solution of the problem? 2-4.

Consider the transient heat conduction in a slab of length l. The slab has the initial temperature distribution T (0, x)

x l (T2 T1 ) T1 l

At both sides of the slab, the following constant temperatures are applied

42

2 Linear Partial Differential Equations

T ( , 0))

1

,

( , ) T2

In addition, the slab contains a heat source. The above given problem can be described by the following partial differential equation (where a and B are constants) ∂T ∂ 2T xBa(( 2 =a 2 + ∂t ∂x l3

1

)

a.) Make the differential equation and the boundary conditions dimensionless by introducing suitable variables. b.) Split the problem into different simpler problems. c.) Solve the different problems and give the complete solution. 2-5.

Consider a sphere with radius R. For 0 the sphere has constant temperature Ti. The surface of the sphere is set to the constant temperature T0 for 0 . The sphere contains also a heat source with constant source intensity qi 0 . The material properties of the sphere are considered to be constant. The temperature distribution in the sphere can be calculated from the energy equation in spherical coordinates

ρc

∂T ∂t

k ∂ § 2 ∂T · k ∂ § T· r si ψ ¨ sin ¸+ ¸+ 2 2 ∂r © ∂r ¹ r sin i ψ ∂ψ © ψ¹ +

k ∂ § T· ¨ ¸ + qi 0 r sin ψ ∂ϕ © ϕ ¹ 2

a.) Simplify the above given energy equation for the case of rotational symmetry. What are the needed boundary conditions? b.) Transform the problem under consideration by using T (r , t ) U (r , t ) / r . What is the resulting differential equation and what are the boundary conditions? c.) Introduce dimensionless quantities, so that the spatial boundary conditions are homogeneous. d.) Solve the transformed problem by using the method of separation of variables. e.) Derive from the solution given under d.) the temperature distribution T( , ).

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

The method of separation of variables has been briefly explained in the last chapter by solving two simple examples. In general, this solution method leads to eigenvalue problems which have to be solved either analytically or numerically. For most technical problems, where analytical solutions based on the method of separation of variables are still obtainable, these eigenvalue problems might become quite difficult. In order to show how the method can be applied to more complicated problems, the present chapter focuses on the solution of heat transfer problems in pipe and channel flows with hydrodynamically fully developed velocity profiles. In Chap. 3, we will also explore some of the general properties of the eigenvalue problems under consideration.

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature As a first example, we investigate the technical important and scientific very interesting problem of convective heat transfer in a turbulent flow of a liquid, flowing in a pipe or a planar channel. We assume that the flow is hydrodynamically fully developed and that the fluid properties are constant. Due to their technical and scientific relevance, these types of flows have been investigated in great detail in the past. Good reviews can be found in Tietjens (1970), Schlichting (1982) and in Bhatti and Shah (1987). Heat transfer problems in laminar pipe or channel flows are also of great theoretical and practical interest. Due to the simplicity of the velocity profile for the hydrodynamically fully developed flow, the heat transfer characteristics for the thermal entry length have been investigated relatively early by Graetz (1883, 1885) and independently by Nusselt (1910). This is the reason that this kind of problems are sometimes referred in literature as Graetz problems. For the heat transfer characteristics in laminar pipe and channel flows, a large number of publications does exist. Good reviews can be found in Shah and London (1978) and in Bhatti and Shaw (1987). In particular, with respect to the thermal development of a hydrodynamically fully developed laminar flow, several theoretical and numerical methods have been evaluated and compared. The reader is referred to Shah and London (1978) for more details. On the other hand, the thermal entrance of a hy-

44

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

drodynamically fully developed turbulent pipe or channel flow has a much broader technical relevance and has not yet been investigated in such detail in the past. Furthermore, this kind of application leads to complicated eigenvalue problems and can be considered as a challenging case for any analytical method. Of course, the present analysis holds also for laminar pipe and channel flows, as stressed in the following sections when relevant. In addition, the program for the solution of the eigenvalue problems, described in Appendix C, can be used for both laminar and turbulent flows. 3.1.1 Velocity Distribution of Hydrodynamically Fully Developed Pipe and Channel Flows Figure 3.1 shows the geometry and the coordinate system of the problem under consideration. We assume a hydrodynamically fully developed flow, which means that the velocity distribution does not change with increasing values of x. This has the implication that only the velocity component u in the x-direction is nonzero. Furthermore, u is only a function of the coordinate orthogonal to the flow direction. This coordinate is y in case of a planar channel and r in case of a circular pipe. In the following the pipe radius is denoted by R and the distance between the two parallel plates by 2h.

Fig. 3.1: Geometry and coordinate system

Furthermore, we assume that the fluid properties are constant. This is a suitable assumption if the temperature differences in the problem are not to large. Then the velocity distribution for the pipe and channel flow can be calculated analytically from the momentum equations. A detailed description is given in Appendix A. For a laminar flow, very simple expressions are obtained: Laminar Pipe Flow

§ § r ·2 · u = 2 ¨1− 1 ¸ ¸¸ u © © R¹ ¹

(3.1)

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature

45

Laminar Flow in a Planar Channel 2 u 3§ y· · = ¨ 1− 1− ¸ ¸ u 2 © © h ¹ ¹¸

(3.2)

where u is the mean velocity of the pipe or channel flow. This mean velocity can be calculated from the measured flow rate in the pipe or channel. For a turbulent hydrodynamically fully developed pipe or channel flow, the turbulent shear stress in the momentum equation in the x-direction has to be modelled by using a turbulence model. For pipe and channel flows, this can be done efficiently by using a simple mixing length model (Cebeci and Bradshaw (1984), Cebeci and Chang (1978), Schlichting (1982)). After some algebra, one obtains finally a description of the velocity distribution in the pipe or in the channel of the following form (the reader is referred to Appendix A for a detailed derivation of the Eqs. (3.3-3.4)). Turbulent Pipe Flow

u §r · = f ¨ , Re R R¸ u R © ¹

(3.3)

Turbulent Flow in a Planar Channel

u §y · = f ¨ ,R Re h ¸ u ©h ¹

(3.4)

Obviously, in a turbulent flow the velocity distribution depends on the Reynolds number. Increasing values of the Reynolds number lead to an enhancement of the turbulent mixing within the cross sectional area of the duct and, therefore, to a flatter velocity profile. Figure 3.2 shows predicted and measured velocity profiles for hydrodynamically fully developed turbulent flows in a pipe and in a planar channel, respectively. In Fig. 3.2, the velocity is scaled by its maximum value in the center. The figure on the left hand side shows a calculation for the flow in a planar channel. It can be seen that the predictions using the simple mixing length model agree quite well with the measurements of Laufer (1950). The hydrodynamically fully developed velocity profile in a pipe is depicted on the right hand side of Fig. 3.2. It can be seen that the velocity profile changes its shape by increasing the Reynolds number from 2.3 104 to 3.2 106. In addition, the good agreement between calculations and measurements of Nikuradse (1932) can be observed (see also Appendix A).

46

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

Planar Channel

Pipe

Fig. 3.2: Hydrodynamically fully developed velocity distribution in a pipe and in a planar channel for different Reynolds numbers

3.1.2 Thermal Entrance Solutions for Constant Wall Temperature

Once the velocity profile is known, the energy equation can be analyzed. If we assume an incompressible flow with constant fluid properties and neglect viscous dissipation, the energy equation takes the following form for the hydrodynamically fully developed turbulent flow: Pipe

ρcp u (r )

∂T ∂x

1 ∂ ª § ∂T r k r ∂r ¬ © ∂r

·º c p v 'T ' » ¹¼

∂ ª ∂T k ∂x ¬ ∂x

º c p u ' T '» ¼

(3.5)

Planar Channel

ρcp u ( y )

∂T ∂x

∂ ª T k y¬ y

c p v 'T '

º ¼

∂ ª ∂T k x x

º c p u ' T '» ¼

(3.6)

Figure 3.3 shows the geometry and the boundary conditions. It can be seen that at x = 0 the wall temperature is suddenly increased from T0 to TW. The fluid tempera-

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature

47

ture has a uniform value T0 for x → -∞. Far away from the entrance (x ( → ∞), the fluid temperature will attain asymptotically the uniform wall temperature TW.

Fig. 3.3: Geometry and boundary conditions

For laminar flow ( u ' ' 00, ' ' 0 ),the equations (3.5) or (3.6) are elliptic in nature. This is caused by the second term on the right hand side of these equations, representing the axial heat conduction effect within the flow. For turbulent flow, the nature of the equation depends also on the turbulent heat fluxes. The turbulent heat fluxes − ρ cP v ′T ′ and − ρ cP u ′T ′ have to be modelled. This can be done for example by using a simple eddy viscosity model:

−v ' T ' = ε hy

∂T (planar channel), ∂y

−u u ' T ' = ε hx

−v ' T ' = ε hr

∂T (pipe) ∂r

(3.7)

∂T (planar channel and pipe) ∂xx

(3.8)

where ε hy , ε hr and ε hx are only functions of the coordinate orthogonal to the flow direction. Inserting Eqs. (3.7-3.8) into Eq. (3.5) and Eq. (3.6) results in Pipe

ρcp u ( r )

∂T ∂x

1 ∂ ª r( r ∂r «¬

) ∂∂Tr º» ¼

∂ ª ( ∂x «¬

) ∂∂Tx º»

(3.9)

¼

Planar Channel

ρcp u ( y )

∂T ∂xx

∂ ª ( y¬

)

Tº ∂ ª ( »+ y¼ x

) ∂Tx º» ¼

(3.10)

Introducing the following dimensionless quantities into the above equations, Θ=

T TW x 1 , x = , y T0 TW L Re L Pr

y r uL , r = , Re L = , L L ν

(3.11)

48

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

εhx

ε hy ε hr u , εhhyy = , u = , Pe L ν ν u

ε hx h , εhhr ν

Re L Pr , Pr =

ν a

where the length scale is L = R for pipe flows and L = h for the flow in a planar channel. We obtain Pipe

u ( r )

∂Θ ∂x

1 ∂ r ∂r

ª «r ( ¬

)

∂Θ º ∂r »¼

1 ∂ ª ( Pee 2L ∂x «¬ P

)

∂Θ º ∂x »¼

(3.12)

Planar Channel

u (

)

∂Θ ∂ ª = (+ ∂x ∂y «¬

)

º 1 ∂ ª + 2 ( » y¼ P Pee L ∂x «

)

∂Θ º x »¼

(3.13)

For Peclet numbers PeeL > 100 and a semi-infinite heating length, axial heat conduction in the fluid can be neglected with good accuracy. This means that the second term on the right hand side of the Eqs. (3.12-3.13) can be ignored. This leads to: Pipe

u ( r )

∂Θ ∂x

u (

)

1 ∂ r ∂r

ª «r ( ¬

)

∂Θ º ∂r »¼

(3.14)

º y »¼

(3.15)

Planar Channel

∂Θ ∂ ª = ( ∂x ∂y «¬

)

These two equations can be combined into one relation, if we introduce a vertical coordinate n, which is equal to y for a planar channel and equal to r for pipe flows. This results in u (

)

∂Θ 1 ∂ ª F = F r ( ∂x ∂n «¬

)

∂Θ º ∂n ¼»

(3.16)

where the superscript F specifies the geometry and has to be set to 0 for a planar channel and 1 for pipe flows. If we examine the nature of Eq. (3.16), we see that the equation is parabolic. The boundary conditions for Eq. (3.16) are therefore given by

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature

49

x = 0 : ((0, (0 ) 1 ∂Θ ∂n n = 0

0

n = 1 1: ( ,1)

0

n = 0 :

(3.17)

From Eq. (3.17), it can be seen that the boundary condition T = T0 for x→ -∞ has been moved to x = 0. This means that the process of heating the wall has no influence on the temperature field forr x < 0. This shows nicely the parabolic nature of the equation. In Eq. (3.16) the dimensionless eddy diffusivity for heat εhn appears as an additional unknown. This quantity is related to the eddy viscosity εm through the similarity between heat and momentum transfer. This is normally indicated by introducing a turbulent Prandtl number defined by Prrt =

εm ε = m ε hn εhn

(3.18)

The turbulent Prandtl number would be equal to one, if there would be full similarity between momentum and heat transfer. However, this is not the case. Therefore, the turbulent Prandtl number will, in general, be a function of the Reynolds and molecular Prandtl number as well as of the distance from the wall. In literature, there are many different models for the turbulent Prandtl number. Good reviews on this subject can be found in Reynolds (1975), Jischa (1982), Kays and Crawford (1993) and Kays (1994). For the moment, it is sufficient to know that, for the problems under consideration, Prt is only a function of n . In the following sections, a few models for the turbulent Prandtl number Prt are provided, which can be used for the calculations of the heat transfer in pipe and channel flows. Introducing the turbulent Prandtl number into Eq. (3.16) leads to u (

)

∂Θ 1 ∂ ª F ∂Θ º = F r a2 ( n ) , « ∂x r ∂n ¬ ∂n »¼

2

( )

1+

Pr εm Prrt

(3.19)

The energy equation (3.19) has to be solved together with the associated boundary conditions Eq. (3.17). This can be done, by using the method of separation of variables. Therefore, we assume that the temperature can be described by Θj = Φj( )

j

(3.20)

( )

Introducing Eq. (3.20) into Eq. (3.19) results in

(

)′ = G ′ ( j

(( ) r F u(

j

( )

)

Gj ( )

(3.21) = Cj

50

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

From this equation, we first solve the ordinary differential equation for the function G j ( ) : G ′j ( )

(3.22)

= Cj

Gj ( ) This equation has the general solution

G j ( x)

D j expp (

)

(3.23)

From Eq. (3.23) it is obvious that the unknown constant Cj must be smaller than zero, since, for Cj = 0, the dependence of the solution on x would be lost and, for Cj > 0, the temperature would tend to infinity for x → ∞ . If we introduce C j = −λ 2j into Eq. (3.23), we obtain G j ( x)

D j expp

(

)

(3.24)

For the function Φj the following ordinary differential equation has to be solved

r F u(( )

2 j

j

( ) ª¬

F 2

( ) ′j ( ) º¼

′

(3.25)

0

Inserting Eq. (3.20) into the boundary conditions Eq. (3.17) results in

n 0 :

∂Θ ∂n n =0

0 Ÿ G(

n 11: Θ ( ,1) 0

) ( )

( ) (1)

( ) 0

(1)

0

(3.26)

0

The differential equation (3.25) has to be solved together with the homogeneous boundary conditions (3.26). As we have seen in Chap. 2 for the elementary examples, non-trivial solutions for this problem exist only for selected values of λ j2 , which are the eigenvalues of Eq. (3.25). Thus, the problem of solving the differential equation (3.25) with boundary conditions (3.26) has been reduced to an eigenvalue problem and is known in literature as a Sturm-Liouville problem (see for example Reid (1980), Kamke (1983), Collatz (1981), Courant and Hilbert (1991) and Sauer and Szabo (1969)). This sort of self-adjoint eigenvalue problem is very common in a lot of physical applications and has been investigated in great detail by several researchers in the past. In the following sections, a short summary of some of the most important properties of this system is given. Properties of the Sturm-Liouville System

In order to show the basic properties of the Sturm-Liouville system, we consider the problem given by the Eqs. (3.25-3.26) in the following general form:

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature

L ª¬

j

λ j M ª¬

º¼

j

º¼

51

(3.27)

with the boundary conditions for the function Φj given by Eq. (3.26). For the eigenvalue problem under consideration, the operators L and M take the following form L ª¬

j

d ª F dΦ j º « r a2 » dn ¬ ddn ¼

º¼ ª¬

j

º¼

(3.28)

( )Φ j

F

(3.29)

The most important properties of the Sturm-Liouville system are: 1. The Problem is Self-Adjoint and Positive Definite

In order to show that the eigenvalue problem under consideration (Eqs. (3.253.26)). is self-adjoint and positive definite, let us define the following two inner products: 1

³ v L [ w] dn

(3.30)

³ v M [ w] dn

(3.31)

< v, w

0

and 1

( , )

0

where v and w are two “compare-functions”. These functions are assumed to be continuously differentiable functions of n in the interval [0,1]. They satisfy the boundary conditions of the problem given by Eq. (3.26) without vanishing in the whole interval. The eigenvalue problem is now called self-adjoint, if < v,,

,

,

( , )

(3.32)

( , )

are satisfied. This means that the inner products, defined by Eqs. (3.30-3.31) are symmetric. Additionally, the eigenvalue problem is called positive definite, if < v,,

0,

( , )

(3.33)

0

Let us first consider the case that the problem is self-adjoint, then 1

< v,,

³ 0

ª¬

F 2

(n) º¼′

1

³ 0

ª¬

F

must be satisfied. Rearranging Eq. (3.34) gives

2

(n) ′º¼′

,v >

(3.34)

52

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems) 1

­

³ ®¯v ª¬ r

F

0

½ w ª¬ r F a2 ( ) ¼ ¾ ¿

a2 ( ) w º¼

(3.35)

0

The expression in the integral in Eq. (3.35) can be rewritten according to v ª¬ r F a2 ( ) w º¼′

w ª¬ r F a2 (n))vv º¼′

ªv ( ¬

)

)º¼′

(

(3.36)

Inserting Eq. (3.36) into Eq. (3.35) and evaluating the inegral results in 0 = v (r a w ) − w(r a v 1

0

)

1 0

= vw′ − vw′ 0 − wv′ + wv′ 0 1

1

(3.37)

From the given boundary conditions, Eq. (3.26), we obtain n

0 : ′j

n 1:

j

0

′ (0)

0,

1

(1)

0,

(0) (1)

(3.38)

0 0

From Eq. (3.38), we see that Eq. (3.37) is identically zero, thus showing that < v, w > = < w, v > is satisfied. The second requirement is that ( , ) 1

³v(

( , )

( , ) . This is also satisfied as

1

)

³ (

0

) ddn

(3.39)

( , )

0

From the preceding analysis, it has been shown that the eigenvalue problem under consideration is self-adjoint. We want now to show that the problem is also positive definite. Therefore, we have to prove that 1

( , )

³v(

1

)

³ (

0

(3.40)

) ddn > 0

0

This is easy to show, since the expression

(

)

is a function ≥ 0 in the interval

under consideration. For the second inner product, we have to show that 1

³

< v,,

0

ª¬

F 2

′ 'º¼ d

(3.41)

0

This can be shown by partial integration of Eq. (3.41). From 1

− ³ v ª¬ r F a2 v 'º¼ dn N

0 a

one obtains

b'

1

ª¬ r a v 'º¼ dn ³ vN'    

F

2

0 a'

b

v ª¬ r F a2 v 'º¼

1 0

(3.42)

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature

53

1

³

< v,,

F 2

( ') 2

(3.43)

0

0

From Eq. (3.43) it is obvious that the expression is larger than zero because r F a2 ( ) is always ≥ 0. Therefore, it has been shown that the eigenvalue problem under consideration is self-adjoint and positive definite. For other wall boundary conditions (constant wall heat flux or boundary conditions of the third kind), the proof that the eigenvalue problem is self-adjoint and positive definite is analogous. 2. Eigenvalues and Eigenfunctions

Having shown that the eigenvalue problem under consideration is self-adjoint, it is relatively simple to prove that the resulting eigenfunctions are orthogonal with respect to a weighting function. In order to show this, the eigenvalue problem (3.27) for two different eigenfunctions Φi and Φj is considered

L ª¬ L[

º¼ λ j M ª¬ ] = λi M [

j

j

]

º¼

(3.44)

Multiplying the first equation by Φi and the second by Φj gives, after subtraction

Φ i L ª¬

j

º¼ Φ j L [ Φ i ] λ j

M ª¬

i

j

º¼ λi Φ j M [

i

]

(3.45)

Integrating both sides of Eq. (3.45) between zero and one gives

< Φ j , Φi > − < Φi , Φ j > = (

1

i

j

)³Φ

i

0

M ª¬

j

º¼ dn

(3.46)

Because the problem under consideration is self-adjoint, the left hand side of Eq. (3.46) is identical to zero. Thus 1

(

)³Φ 0

i

M ª¬

j

º¼ dn = 0

(3.47)

From this equation, it is obvious that for i = j, the left hand side of Eq. (3.47) is identical to zero, because the eigenvalues are the same. For i ≠ j, the integral 1

³ Φ M ª¬ i

0

j

º¼ dn has to be zero in order to fulfill Eq. (3.47). Inserting the definition

of the operator M into Eq. (3.47) results in 1

³r 0

F

u( )

i

j

0 for i ≠ j

(3.48)

54

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

This means that the eigenfunctions are a set of orthogonal functions in the interval [0,1] with respect to the weighting function ( ) defined by F

g( )

(3.49)

( )

Eq. (3.48) is an important result for the solution of the partial differential equation, as it can be used later to expand an arbitrary function in terms of eigenfunctions. 3. The Eigenvalues of the Sturm-Liouville System

Normally, the eigenvalues of Eq. (3.27) can only be obtained numerically. Therefore, it is important to find out if we can restrict in advance the search interval for these values. Let us assume that one eigenvalue is complex λ j α + i β with the eigenfunction Φ j = a ib . Then its complex conjugate λi

α − i β is also an ei-

genvalue with the eigenfunction Φ i = a ib . If we introduce these values into Eq. (3.47), one obtains 1

2 β³

F

(

b )d

(3.50)

0

0

F Since the weighting function ( ) ( ) is larger or equal to zero in the investigated interval, the integral in Eq. (3.50) must be bigger than zero. Therefore, in order to satisfy relation (3.50), β must be equal to zero. This shows that the eigenvalue problem has only real eigenvalue. Since the eigenvalue problem is also positive definite, it can be shown (see Collatz (1981)) that all eigenvalues are larger or equal to zero. This shows the correctness of our previous assumption C j = −λ 2j , which was based only on physical reasoning.

4. Eigenfunction Expansions for an Arbitrary Function

We have already shown that the eigenfunctions form an orthogonal set of functions. Now we want to prove that an arbitrary “well behaved” function ( ) can be represented by an infinite number of eigenfunctions. We assume, therefore, that ∞

f

( ) ¦

j

j =0

j

(3.51)

( )

is an uniformly convergent series. Multiplying both sides of Eq. (3.51) by g ( ) i ( ) and integrating over n between zero and one leads to 1 ∞

1

³ 0

f ( )g ( )

i

( )

³¦ 0 j =0

j

g ( n)

i

j

dn d

(3.52)

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature

55

Interchanging on the right hand side of the equation summation and integration results in ∞

1

³

f ( )g( )

1

¦ ³

i( )

j

j 0

0

( )

i

j

dn d

(3.53)

0

From Eq. (3.53) and Eq. (3.48) it can be seen that the integral on the right hand side is equal to zero if i ≠ j. Therefore, the constants Aj are given by 1

³ g ( n) f ( n)

Aj =

j

(n)ddn

0

(3.54)

1

³ g( )

2 j

( ) ddn

0

The present investigation is intended only to show some of the important properties of the Sturm-Liouville system. The reader is referred to Coddington and Levinson (1955), Sagan (1989), Reid (1980), Kamke (1977) and Myint-U and Debnath (1987) for a more detailed and rigorous mathematical treatment of the SturmLiouville eigenvalue problems. From the previous analysis, it is clear that the solution of the partial differential equation (3.19) can be constructed by linear superposition of solutions given by Eq. (3.20). This results in the following expression for the temperature distribution in the fluid ∞

Θ = ¦ Aj Φ j ( ) exp j =0

(

)

(3.55)

The constants Aj in Eq. (3.55) can be obtained by satisfying the boundary condition for 0 . Here we have ∞

1

¦

j

j

( )

(3.56)

j =0

From Eq. (3.54) it is obvious that the constants Aj in Eq. (3.56) are given by 1

Aj =

³r

F

³r

F

u

j

ddn

u

2 j

ddn

(3.57)

0 1

0

The numerator of Eq. (3.57) can be further simplified, by using the differential equation for the eigenvalue system given by Eq. (3.25). Replacing the expression in the integral by Eq. (3.25) leads to

56

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems) 1

F ³r u 0

1

j

dn

1

³

λj

0

=−

1

λj

2

2

′ ª¬ r F a 2 ( n ) ′j ( n ) º¼ dn d =

ª r F a2

′j º

1 0

=−

1

λ j2

(3.58)

Φ ′j (1)

where a2 (1) 1 has been used. Thus, the constants Aj are finally given by Aj =

−Φ′j (1)

(3.59)

1

λ j2 ³ r F u

2 j

dn d

0

Combining Eqs. (3.55) and (3.59), the temperature distribution in the fluid is completely known. In Chap. 2, we have seen how the eigenvalues and eigenfunctions can be predicted analytically. However, for most technical relevant cases, the eigenvalues and eigenfunctions can only be predicted numerically. In Appendix C a FORTRAN program is described which can be used for this purpose. However, it should be noted, that larger eigenvalues (j (j > 10) might be predictable by asymptotic formulas. This is explained in detail in Chap. 4. After the temperature distribution in the fluid is known, the Nusselt number can be calculated. The Nusselt number is defined as: Nu D =

hD k

(3.60)

In Eq. (3.60), the Nusselt number depends on the heat transfer coefficient h and the hydraulic diameter D. The hydraulic diameter D is defined as D=

4A U

(3.61)

where A is the cross sectional flow area of the duct and U is the wetted perimeter. For a circular pipe, one obtains D = 2R. For a rectangular channel, the geometrical parameters are shown in Fig. 3.4.

Fig. 3.4: Geometry of a rectangular channel

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature

57

The channel has width W and height 2h. Following Eq. (3.61), the hydraulic diameter is equal to D=

4 (2 ) 4h = 4h 2W 2h + 1 W

(3.62)

For the here considered case of a two-dimensional channel (h/W → 0), Eq. (3.62) leads to D =4 h. The heat transfer coefficient h, which appears in Eq. (3.60) can be obtained from an energy balance at the surface where the heat transfer process occurs. Here we have h (T

)

T

k

∂T ∂n

(3.63) n L

Replacing the heat transfer coefficient h in Eq. (3.60) with Eq. (3.63), we obtain the following definition for the Nusselt number

D Nu D =

∂T ∂n

TW

(3.64) n L

Tb

From this definition, it is obvious that the Nusselt number is a dimensionless temperature gradient at the wall. In Eq. (3.64), TW denotes the wall temperature, while Tb is the “bulk-temperature” defined by L

Tb =

³ uuTr

F

dn

(3.65)

0 L

³ ur

F

dn

0

The bulk-temperature is a mass averaged fluid temperature. Introducing the dimensionless quantities defined by Eq. (3.11), we obtain the Nusselt number and the bulk-temperature in dimensionless form −4 Nu D =

∂Θ ∂ n n =1

(3.66)

2 F Θb 1

Θb = 2 F ³ ur F Θ ddn

(3.67)

0

Introducing now the known temperature distribution in the fluid (Eq. (3.55)) results in the following expression for the bulk-temperature

58

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

Φ ′j ((1))

∞

Θ b = −2 F ¦

j

j =0

λj

2

exp

(

)

(3.68)

and for the Nusselt number ∞

4¦

j

j =0

Nu D =

4

F

Φ ′j ((1))

∞

¦ j =0

′j (1) exp

j

λ j2

(

exp

) (

(3.69)

)

where F = 0 denotes the heat transfer in a planar channel whereas F = 1 denotes the heat transfer in a circular pipe. If we are only interested in the Nusselt number for the hydrodynamic and thermal fully developed flow, we can obtain the Nusselt number for the fully developed flow from Eq. (3.69) by setting x → ∞ . For this limit only the first term of the sums has to be retained because of the very fast growing eigenvalues with increasing j. One obtains therefore the very simple expression lim Nu D x →∞

Nu ∞ =

4 2 λ0 4F

(3.70)

We have obtained the result that, in case of a fully developed flow, the Nusselt number (for the constant wall temperature boundary condition) depends solely on the first eigenvalue. This result is important for turbulent flows where the thermal entrance length is normally quite short due to the rapid mixing in the flow. Laminar Flows

As stated earlier, the heat transfer process, in hydrodynamically fully developed laminar flow in a planar channel and in a pipe, has been extensively studied in literature (see Shah and London (1978)). Therefore, only some results are reported here which help to understand the general behaviour of the heat transfer process in laminar duct flows. One of the most important results for this type of problems is that the temperature distribution as well as the distribution of Nusselt number depend only on the dimensionless axial coordinate x =

x 1 L Re L Pr

(3.71)

and not explicitely on the Reynolds- or Prandtl number (for a pipe flow L = R and for a flow in a planar channel L = h). This can easily be understood, if we examine Eq. (3.19). For laminar flows, the velocity distribution is only a function of n and not a function of the Reynolds number (see Eqs. (3.1-3.2)). Additionally, the function a2 ( ) =1. Therefore, the problem given by Eq. (3.19) and Eq. (3.17) does not depend on the Reynolds- and the Prandtl number. This fact leads to great simplifi-

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature

59

cations for these type of problems, because the eigenvalues and eigenfunctions have only to be calculated once. The qualitative distribution of the bulktemperature is displayed in Fig. 3.5, whereas the qualitative distribution of the Nusselt number is given in Fig. 3.6. From Fig. 3.5, it can be seen that the bulktemperature at the inlet of the duct is equal to the uniform inlet temperature T0. This is obvious from the definition of Tb, given by Eq. (3.65). With increasing values of x , the bulk-temperature attains asymptotically the uniform wall temperature TW. The distribution of the Nusselt number is shown in Fig. 3.6. Here it can be seen that the value of the Nusselt number is extremely high for very small values of the x coordinate. This is caused by the temperature jump in the wall temperature at the inlet of the duct, which leads to the development of the temperature boundary layer at 0 . With increasing values of the axial coordinate, the Nusselt number decreases and reaches an asymptotic constant value. The value of this constant,

Fig. 3.5: Distribution of the bulk-temperature for a laminar, hydrodynamically fully developed pipe flow with constant wall temperature

Fig. 3.6: Distribution of the Nusselt number for a laminar, hydrodynamically fully developed pipe flow with constant wall temperature

60

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

however, depends on the geometry and applied wall boundary conditions. For the present case of constant wall temperature, the value for the fully developed Nusselt number is Nu

3.657

(3.72)

pipe

Nu 7.5407 planar channel Having shown that, for hydrodynamically fully developed flows, both the temperature field and the distribution of the Nusselt number depend only on the axial coordinate x , simple correlations can be obtained quite easily for calculating the Nusselt number and, thus, the heat transfer in ducts for laminar flow. For example, the distribution of the Nusselt number in a pipe for the thermal entrance region can be approximated by (see Shah and London (1978)): ­ 1.077 − 0.7 , * 0.01 13 ° °( ) Nu D = ® °3.657 + 6.874 exp( 57.2 *)) , * 0.01 0.488 0 488 ° ( ) ¯

(3.73)

where a slightly modified axial coordinate has been used: x* =

(3.74)

x 1 D Re D Pr

D is the hydraulic diameter ((D = 2R for the pipe and D = 4h for the planar channel). For the heat transfer in a planar duct similar expressions can be obtained (see Shah and London (1978)): ­ 1.233 − 0.4 13 ° °( ) Nu D = ® °7.541 + 6.87400.488 exp( 245 *)) 488 ° ( ) ¯

, * 0.001

(3.75)

, * 0.001

It is interesting to note that the dependence on the axial coordinate is quite similar for both solutions (Eqs. (3.73) and (3.75)). Only the matching constants differ in order to approximate the heat transfer in a pipe or in a planar channel. Turbulent Flows

Heat transfer in turbulent duct flows is of greater technical relevance than laminar flow. In turbulent flows, the mixing process is much more effective than in laminar flows. This produces a shorter hydrodynamic entrance length compared to laminar duct flows, while the velocity profiles are much flatter. Thanks to the shorter hydrodynamic entrance length, the flow can be described for a lot of applications either as hydrodynamically fully developed or as hydrodynamically and

3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature

61

thermally fully developed. Exceptions are represented by turbulent flows of liquid metals. Due to the very low Prandtl number for these types of applications, the heat transfer behavior of such flows might be more similar to laminar flows. Before discussing some heat transfer results, the turbulent Prandtl number ( Prrt m / ε h ) has to be specified. As it can be seen in Eq. (3.19), the turbulent Prandtl number will significantly influence the term a2 (

) . If we assume that the

transfer of momentum and heat is similar we can set Prt =1. Unfortunately, this simple approximation is not realistic for many applications. In literature, a lot of different approaches are described to model the turbulent Prandtl number. The reader is referred to Reynolds (1975), Jischa (1982), Kays and Crawford (1993) and Kays (1994) for good reviews and comparisons between different models. In general it can be noted that for Re D ≠ 0

Prrt Prrt

const. for Re D

and Pr

0

and Pr

1

(3.76)

This can be explained in the following way: for Pr 0 ). Note that if there is a jump in the wall temperature distribution, the solution is then given by Eq. (3.55) and has only to be added to the solution obtained here. The problem under consideration is shown in Fig. 3.11.

Fig. 3.11: Duct with arbitrary wall temperature

If we introduce a new dimensionless temperature Θ1 , defined by Θ1 =

( ) TW ( )

(3.90)

T TW T0

the energy equation becomes u (

)

∂Θ1 1 ∂ ª F = F r ( ∂x ∂n «¬

)

∂Θ1 º u ( ∂n »¼

)

d ΘW ddx

(3.91)

0

(3.92)

with the boundary conditions x = 0 : n = 0 :

1

( 0, )

∂Θ1 ∂n

1, temperature jump at 0

n = 0

n = 1 1: Θ1 ( ,1) = 0 and the abbreviation ΘW = (

( )) / (

( )) .

As it can be seen from the boundary conditions, Eq. (3.92), and also from Fig. 3.11, the wall temperature distribution has a jump at x = 0 . Introducing the new defined temperature Θ1 into the energy equation results in a new term on the right hand side of Eq. (3.91). On the other hand, the boundary conditions for n = 0 and n = 1 are homogenous for Θ1 . This means that the solution for the case of an arbitrary wall temperature requires the solution of the non-homogenous equation. In the previous section we have seen that an arbitrary function could be represented by an infinite number of eigenfunctions. Thus, the following expression for the temperature distribution is assumed

68

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

∞

Θ1 = ¦ Bm ( m =0

)

(3.93)

( )

The eigenfunctions in Eq. (3.93) are already known from the solution of the eigenvalue problem (see Eqs. (3.25-3.26)). In the following analysis, we try to find a solution of the above non-homogeneous problem by using the orthogonality properties of the eigenfunctions. Multiplying both sides of Eq. (3.93) by u r F Φ n ( ) and integrating the resulting expressions between 0 and 1 gives the following equation for the functions Bm 1

Bm =

³u r

F m

(3.94)

Θ1 ddn

1

=

0

1

³u r

F

2 m

dn

1 u rF K m ³0

m

Θ1 dn d

0

Because of the boundary condition for Θ1 for isfy the following boundary condition Bm (

)

1 Km

1

1 m

0

1

Km

0 , the functions Bm have to sat-

1

³ 1 ª¬ r 0

F

−Φ ′m ( ) a2 ′m º¼′ ddn = 2 λm K m

(3.95)

The two other boundary conditions for n = 0 and n = 1 are automatically satisfied by Eq. (3.93) because of the employed eigenfunctions. If we differentiate Eq. (3.94) with respect to x , we obtain an ordinary differential equation for the unknown functions Bm 1 dBm 1 = ur F dx K m ³0

m

∂Θ1 ddn ∂∂xx

(3.96)

∂ 1 / ∂x in the above equation can be replaced by using the enThe expression u ∂Θ ergy equation (3.91). This results in 1 dBm 1 ∂ ª F = r a2 ( n ) 1 » dx K m ³0 ∂n « x

m dn

1 K

1

³ ur 0

F m

d ΘW ddn dx

(3.97)

The first integral on the right hand side of Eq. (3.97) can be solved by partial integration and taking into account the boundary conditions for Θ1 and Φ m . The second integral can be solved directly by using the differential equation for the eigenfunctions, Eq. (3.25). This results in Φ ′ ( ) d ΘW dBm = −λm2 Bm − 2m dx λm K m ddx The ordinary differential equation (3.98) has the general solution

(3.98)

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux

Bm = Cm exp e p(

)

( ) x d ΘW

Φ′

³

λ Km 2 m

0

dx

expp (

(

) ) dξ

69

(3.99)

The constants Cm can be determined by comparing Eq. (3.99) with Eq. (3.95) for x = 0 . This results in Bm = −

Φ′m ( ) ­° p( ®eexp λm2 K m °¯

)

m

x

0

d

W

ddx

eexp p(

(

½

) ) dξ °¾

(3.100)

°¿

Introducing Eq. (3.100) into Eq. (3.93) results in the following equation for the temperature distribution Θ1 = Θ (

x

) ³ (( 0

In this equation, Θ (

)

) )

d ΘW dξ ddx

(3.101)

denotes the temperature distribution for the constant

wall temperature solution. For W const. Eq. (3.101) states that Θ1 = Θ . From Eq. (3.101) it is obvious, that the temperature distribution for the case of an arbitrary wall temperature can be constructed if the temperature distribution for the uniform wall temperature case is known. This shows very nicely the power of the superposition approach. It should be noted here, that the solution according to Eq. (3.101) can also be obtained by other methods, for example, by using Duhamel’s theorem. The reader is referred to Kays and Crawford (1993) for a different method to derive the above solution for the temperature field for the axial varying wall temperature. The present solution has the advantage that the solution method can also be used in the same way for an arbitrary source term in the energy equation.

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux Fluid flow and heat transfer in rotating systems are not only of considerable theoretical interest, but also of great practical importance. Therefore, transport phenomena in rotating systems have challenged engineers and scientists for a long time. For example, Lord Rayleigh (1917) investigated the dynamics and stability of revolving fluids. Also, some of the classical solutions of the Navier-Stokes equations were obtained for rotating systems (see Schlichting (1982)). Von Karman (1921) investigated the flow induced by a rotating disk and the associated convective heat transfer. The stability of a circular flow in an annulus, formed between two concentric rotating cylinders, was studied by Taylor (1923). The turbulent flow and heat transfer in an axially rotating pipe is a rather elementary configuration and is therefore suitable as a test case for new turbulence

70

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

models. Furthermore, many technical applications in rotating machinery are using this simple type of geometry. One of these technical applications is the cooling of gas turbine shafts by means of air flowing through a longitudinal hole in the shaft itself. When a fluid enters a pipe revolving around its axis, tangential forces acting between the rotating pipe and the fluid cause the fluid to rotate with the pipe, resulting in a flow pattern rather different from the one observed in a non-rotating pipe. Rotation was found to have a remarkable influence on the suppression of the turbulent motion because of radially growing centrifugal forces. The effect of pipe rotation on the hydraulic loss has been investigated experimentally by Levy (1929), White (1964) and Shchukin (1967). If the flow is turbulent, the flow is stabilized with increasing rotation rate and the turbulence is suppressed by the centrifugal forces. This can be seen in Fig. 3.12, where flow visualization results from White (1964) are shown. For turbulent flow in a rotating pipe, Borisenko et al. (1973) studied the effect of rotation on the turbulent velocity fluctuations using hot-wire probes and showed that they were suppressed.

ReeD = 3520, Re = 0: Turbulent Flow

ReeD = 3520, Re = 4800: Laminarized Flow Fig. 3.12: Flow visualization experiments for the flow in an axially rotating pipe (White (1964))

Murakami and Kikuyama (1980) measured the time-mean velocity components and hydraulic losses in an axially rotating pipe when a fully developed turbulent flow was introduced into the pipe. The results were obtained in dependence of the Reynolds number Re D /ν , the rotation rate N = wW / u and the length of the rotating pipe. The rotation was found to suppress the turbulence in the flow, and also to reduce the hydraulic loss. With increasing rotational speed, the axial velocity distribution finally approaches the Hagen-Poiseuille flow. Kikuyama et al. (1983) calculated, for the case of a hydrodynamically fully developed flow, the

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux

71

distribution of the axial velocity component by using a simple mixing length approach according to Bradshaw (1969). The distribution of the tangential velocity component was approximated by w §r· =¨ ¸ wW © R ¹

2

(3.102)

which is in good agreement with experimental data from Murakami and Kikuyama (1980), Kikuyama et al. (1983) and Reich and Beer (1989). Reich and Beer (1989) extended the mixing length approach of Kikuyama et al. (1983) and calculated, for the case of a hydrodynamic and thermal fully developed flow, the axial velocity component and the temperature distribution in the fluid. They showed that in agreement with their measurements, the heat transfer decreased with increasing rotation rate. Hirai and Takagi (1987) and Hirai et al. (1988) calculated the velocity distribution and the heat transfer for the fully developed flow in the axial rotating pipe. In their calculations, it has not been necessary to prescribe the tangential velocity distribution (like Eq. (3.102)) at the beginning of the calculation procedure as it has been done by Kikuyama et al. (1983) and Reich and Beer (1989). However, they obtained the universal distribution of the tangential velocity as a result of their calculations. For the hydrodynamically fully developed flow Weigand and Beer (1989a) solved analytically the energy equation and calculated the local distribution of the Nusselt number for the thermal entrance region. The calculated Nusselt numbers compared very well with the experimentally obtained results of Reich (1988). For the case of a convectively cooled pipe, Weigand and Beer (1992b) obtained an analytical solution for the Nusselt number distribution in the thermal entrance region. It could be shown, that the thermal wall boundary condition influenced the values of the Nusselt number. This is because of the laminarization effect within the pipe. For the case of a hydrodynamically and thermally developed flow, Weigand and Beer (1992a) calculated numerically the distribution of the axial velocity distribution and the local Nusselt number as a function of the axial coordinate. Because of the simple geometry involved, the fully developed flow in an axial rotating pipe has been investigated both by using LES (large eddy simulation) and DNS (direct numerical simulation). Eggels and Nieuwstadt (1993) calculated the axial and tangential velocity distribution in the axial rotating pipe by using LES. Their calculations showed the parabolic distribution of the tangential velocity component and agreed well with measurements of Reich (1988) and Nishibori et al. (1987). Orlandi and Fatica (1997) investigated the flow by DNS for low Reynolds numbers. Malin and Younis (1997) employed the closures for the transport equations of Gibson and Younis (1986) and Speziale et al. (1991) to predict the flow and heat transfer in an axially rotating pipe. The results were compared to experimental data and LES results and good agreement was found for both models. Rinck and Beer (1999) used a modified Reynolds stress model to predict the fully developed flow in an axially rotating pipe. They found good agreement between their predictions and experimental data of Reich (1988). Recently, Speziale et al. (2000) presented a comprehensive analysis of the modeling of turbulent flows in rotating pipes. They discussed also the main physi-

72

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

cal characteristics of the flow and surveyed the capabilities of different classes of closure models to reproduce them. 3.3.1 Velocity Distribution for the Hydrodynamically Fully Developed Flow in an Axial Rotating Pipe

The hydrodynamically fully developed velocity distribution in an axially rotating pipe can be calculated by using a simple mixing length approach. This procedure is only briefly outlined here, since our main focus is on analytical methods relevant for solving heat transfer problem. A more detailed explanation of the method as well as a review on other methods for calculating the velocity distribution in an axial rotating pipe is given in Appendix B.

Fig. 3.13: Physical model and coordinate system (Reich and Beer (1989))

Figure 3.13 shows the geometry and the cylindrical coordinate system with the x, r, ϕ axis and the related velocity components u, v, w. If we assume a hydrodynamically fully developed turbulent flow with constant fluid properties, the Navier-Stokes equation for the rotational symmetric case are given by (see for example Reich (1988)): −ρ

) + ρ wrw

(3.103)

∂ § w· · 2 ¸ ρr ′ ′ ¸ ∂r © r ¹ ¹

(3.104)

w2 ∂p ρ ∂ =− − r ∂r r ∂r

(

0

1 ∂ § ¨μ r 2 ∂r ©

0=

∂p 1 ∂ § ∂u + ¨ μr ∂x r ∂r © ∂r

3

′ ′

· ¹

ρ ru v ¸

(3.105)

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux

73

with the boundary conditions r

0:

r

R:

00,

0, 0 W

,

∂u ∂r 00,

(3.106)

0 0

Since we assume a hydrodynamically fully developed flow, the radial velocity component v is zero everywhere and the continuity equation results in the fact that the axial velocity component u is only a function of r. As mentioned before, experimental data indicate that the tangential velocity distribution is universal and can be approximated by Eq. (3.102). Fig. 3.14 shows the tangential velocity distribution for different rotation rates N as well as for different Reynolds numbers. From Fig. 3.14 it is obvious that Eq. (3.102) is a very good approximation for the tangential velocity distribution. Introducing t Eq. (3.102) into Eq. (3.104) shows that v′w′ C r . This relation is of course only correct far away from the wall. However, it is interesting to note that the above given linear relation has been confirmed by DNS calculations.

Fig. 3.14: Tangential velocity distribution (Reich and Beer (1989))

The axial velocity distribution u(r) can be calculated from Eq. (3.105). In order to do this, the turbulent shear stress in this equation has to be related to the mean velocity gradients. This can be done by using a mixing length model according to Koosinlin et al. (1975). This results in the following expression for the turbulent shear stress ª § ∂u · 2 § ∂ § w · · 2 º ρ u ′v ′ = ρ l « ¨ ¸ + ¨ r ¨ ¸¸ » «¬ © ∂rr ¹ © ∂ r © r ¹ ¹ »¼ 2

1/ 2

∂u ∂u = εm ρ ∂r ∂r

where the mixing length distribution l is given by (Reich and Beer (1989))

(3.107)

74

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

§ 1 · = ¨1 Ri ¸ l0 © 6 ¹

l

(3.108)

2

The mixing length distribution l0 is the one for a non-rotating pipe (see Appendix A, Eq. (A.20)). The Richardson number in Eq. (3.108) describes the effect of pipe rotation on the turbulent motion and is defined by w ∂ ( ) r ∂r Ri = 2 § ∂u · § ∂ § w · · ¨ ¸ + r ¨ ¸¸ © ∂rr ¹ © ∂ r © r ¹ ¹

(3.109)

2

2

Without rotation, Ri = 0 and there exists a fully developed turbulent pipe flow. If Ri > 0, i.e. for an axially rotating pipe with a radially growing tangential velocity, the centrifugal forces suppress the turbulent fluctuations and the mixing length decreases. Inserting Eqs. (3.107-3.109) into Eq. (3.105) results in a strongly nonlinear ordinary differential equation for the axial velocity component u. This is shown in detail in Appendix B. Fig. 3.15 shows a comparison between calculated and measured axial velocity distributions according to Reich and Beer (1989). It can be seen that the numerical calculations agree well with the measurements. In Appendix B, an analytical approximation for the velocity distribution in the axial rotating pipe is given (Weigand and Beer (1994)).

Fig. 3.15: Axial velocity distribution as a function of the rotation rate N (Reich and Beer (1989))

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux

75

3.3.2 Thermal Entrance Solution for Constant Wall Heat Flux

The thermal entrance solution for the case of a constant heat flux at the wall can also be obtained by the method of separation of variables. However, because of the different wall boundary condition, the approach for solving this problem is different to the case of a constant wall temperature. If we assume that the flow is incompressible and that the fluid properties are constant, the energy equation for the hydrodynamically fully developed flow, with negligible axial heat conduction effects, takes the form ∂T ∂x

ρcp ( )

1 ∂ ª § ∂T ¨ r ∂r «¬ © ∂ r

ρ

(3.110)

·º T ¸» ¹¼

If we replace the turbulent heat flux in Eq. (3.110) by the eddy viscosity and the turbulent Prandtl number, we obtain ª 1 ∂ r ∂r « ° «¬ ¯

∂T ρc p u ( ) ∂x

p

2 2 l 2 ª ∂u w ¨ ¸ ¨ ¨ ¸¸ Pr t «¬© ∂r ¹ © ∂rr © r ¹ ¹ »¼

1/ 2

º ∂T ¾ r »» °¿ ∂r »¼

(3.111)

with the turbulent Prandtl number for air flow given by 1 Pr t

§r· 1.53 1 53 2.82 ¨ ¸ © R¹

2

§r· 3.85 3 85 ¨ ¸ ©R¹

3

§r· 1.48 1 48 ¨ ¸ © R¹

4

(3.84)

Eq. (3.111) has to be solved with the following boundary conditions (3.112)

x 0 : T T0 r 0:

∂T ∂r

: −k

00,,

∂T = qW = const. ∂r

For the constant wall temperature case, the driving temperature potential TW –T0 can be used to scale the temperature. This is not the case for a constant wall heat flux. Therefore, the following quantities are used to make the equations dimensionless: Θ=

(

T T0

)

, 

r

,

=

x

2 Re D Pr

,

u uD = , R Re D = u ν

(3.113)

As it can be seen from Eq. (3.113), the temperature is made dimensionless with the heat flux density at the wall, while the axial and radial coordinates are made dimensionless in the usual way. Introducing the dimensionless quantities into the Eqs. (3.111-3.112) results in:

 ur

∂Θ ∂x

∂ ª ∂r «¬

2

( )

∂Θ º ∂r »¼

(3.114)

76

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

with the boundary conditions x = 0 : Θ = 0 ∂Θ r 0 : ∂r and the function a2 (

(3.115) ∂Θ 1: ∂r

00,,

1

) 1/ 2

a2 (

2 ª§ ∂u · 2 § 1 Pr · º R D l 2 «¨ ¸ ¨ r N ¸ » Re ) =1 Prrt ¹ »¼ «¬© ∂r ¹ © 2

(3.116)

Because the wall boundary condition for r = 1 is non-homogeneous, the solution for the problem given by Eqs. (3.114-3.115) has to be split into two problems. In the first problem only a particular solution for the energy equation is searched. The second problem can then be solved afterwards by using the method of separation of variables. This can be done, because the energy equation, as well as the wall boundary conditions, are linear. Introducing the function

Θ = Θ1 + Θ2

(3.117)

into the above equations results in the two problems

 ur

∂Θ1 ∂x

∂ ª ∂r «¬

2

∂Θ1 º ∂r »¼

( )

(3.118)

with the boundary conditions r 0 :

∂Θ1 ∂r

0,, 0

 ur

∂Θ 2 ∂x

∂ ª ∂r «¬

1:

∂Θ1 ∂r

( )

∂Θ 2 º ∂r ¼»

1

(3.119)

and 2

(3.120)

with the boundary conditions x = 0 : Θ 2 = − Θ1 r 0 :

∂Θ 2 ∂r

0,, 0

(3.121) 1:

∂Θ 2 ∂r

0

Because only one particular solution of the energy equation has to be found by solving the problem for Θ1 , the boundary condition for 0 can be ignored. This implies that, for the second problem, the boundary condition for 0 has to be modified. Physically, the solution Θ1 represents the temperature distribution for

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux

77

the fully developed flow, i.e. far away from the pipe entrance, where the fluid temperature monotonously increases due to the heat addition through the wall. Temperature Distribution for the Fully Developed Flow ( Θ1 )

For the fully developed flow, the temperature distribution in the pipe increases linearly with growing values of the axial coordinate. This can be shown by looking at an energy balance for the pipe section shown in Fig. 3.16. The energy balance leads to 2π R

2

qW

³ ³ρ

p

(

0

)urdrd d dϕ

(3.122)

0 0

Introducing the dimensionless quantities from Eq. (3.113) into Eq. (3.122) gives 1

x

d ³Θ urdr

(3.123)

1

0

Fig. 3.16: Energy balance for the pipe with an uniform heat flux at the wall

From Eq. (3.123), it is obvious that the temperature in the fluid increases linearly with growing axial distances for the fully developed flow. This motivates the introduction of the following expression for Θ1 Θ1 = C1 x + ψ (

)

(3.124)

Introducing Eq. (3.124) into Eqs. (3.118-3.119) and into Eq. (3.123) results in u r C1

d § dψ · a2 ( r ) r ¸ dr © dr ¹ d

(3.125)

78

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

r

0:

d dr

0,, 0

1: 1

1

x

dψ dr d

(3.126)

1

1

(3.127)

d ³ ψ ( r ) urdr

d ³ x C1urdr 0

0

From Eq. (3.127), it can be seen that the unknown constant C1 is determined by the integral energy balance. Solving the above set of ordinary differential equations results in the following temperature distribution for the fully developed flow Θ1 (

) = −2 x + ( r ) + C2

(3.128)

with the abbreviations

ψ( )

§ ³0 ¨ η © r

η · 2 ( ξ )ξ ξ ¸ ³ ¸ 2( ) 0 ¹

1

,

³ 2ψ (

2

  dr d ) ru

(3.129)

0

The Temperature Distribution Θ 2

Introducing the obtained temperature distribution for the fully developed flow into Eq. (3.121) results in the following problem for Θ 2  ur

∂Θ2 ∂x

∂ ª ∂r «¬

x = 0 : Θ 2 = r 0 :

∂Θ 2 ∂r

2

( )

∂Θ 2 º ∂r »¼

(3.120)

( r ) + C2

(3.130)

0,, 0

1:

∂Θ 2 ∂r

0

The latter can be solved using the method of separation of variables. Introducing Θ 2 j = Fj (

)

j

( )

(3.131)

into Eqs. (3.120, 3.130) results in the following eigenvalue problem

)′

(

λ 2j

′j

0

′j

0

(3.132)

with the boundary conditions r

0:

′j

0, 0

1: 1:

(3.133)

and an arbitrary normalization condition r = 0 :

j

1

(3.134)

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux

The functions Fj (

)

79

are given by Fj ( )

exp( e p(( λ j2 )

j

(3.135)

The eigenfunctions can be predicted with a program similar to the one given in Appendix C. The temperature distribution Θ 2 is then given by ∞

Θ2 = Σ

j =0

j

j

( ) exp (

(3.136)

)

The constants Aj can be calculated with a procedure similar to the one employed in section 3.1 for the constant wall temperature case. Using the orthogonality of the eigenfunctions leads for the present case to 1

Aj = ³ ( ) ( )

(3.137)

1

j

( )

0

/³

2 j

( )ddr

0

Both the eigenvalues and eigenfunctions can be calculated numerically. Table 2.1 reports some eigenvalues and constants Aj Φ j ( ) calculated for different Reynolds numbers and rotation rates N. Note that the eigenvalues increase with increasing Reynolds number and decrease with increasing rotation rate. This means that increasing Reynolds numbers result in a better mixing in the flow and lead to shorter thermal entrance lengths, whereas increasing rotation rates lead to laminarization and therefore to a longer thermal entrance length. Table 2.1. Eigenvalues and constants for different rotation rates and different Reynolds numbers (Pr = 0.71).

ReD = 5000 N

λ02

λ12

λ22

A0 Φ 0 (1)

A1Φ1 (1)

A2 Φ 2 (1)

0 1 2 3 5

0 0 0 0 0

152.95 80.76 46.13 30.56 18.62

436.18 250.74 146.25 97.52 59.73

0 0 0 0 0

0.0287 0.0409 0.0654 0.0946 0.1467

0.0156 0.0183 0.0259 0.0358 0.0537

N

λ02

λ12

λ22

A0 Φ 0 (1)

A1Φ1 (1)

A2 Φ 2 (1)

0 1 2 3 5

0 0 0 0 0

505.25 216.04 108.93 63.24 28.57

1476.01 687.15 350.64 203.82 92.04

0 0 0 0 0

0.0077 0.0140 0.0266 0.0450 0.0960

0.0044 0.0058 0.0010 0.0165 0.0349

ReD = 20000

80

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

ReD = 50000 N

λ02

λ12

λ22

A0 Φ 0 (1)

A1Φ1 (1)

A2 Φ 2 (1)

0 1 2 3 5

0 0 0 0 0

1115.14 436.34 210.11 115.92 44.17

3281.90 1396.90 676.36 375.17 142.76

0 0 0 0 0

0.0034 0.0068 0.0136 0.0246 0.0630

0.0018 0.0027 0.0048 0.0088 0.0225

From Table 2.1, an interesting fact can be noticed. For the considered wall boundary conditions, λ0 = 0 is a possible eigenvalue. This eigenvalue results in the possible eigenfunction Φ 0 = 1 . If we introduce this eigenvalue and the eigenfunction into Eq. (3.137) we obtain A0 = 0 (because the numerator of Eq. (3.137) is zero according to Eq. (3.127)). Therefore, λ0 = 0 can be excluded from the considerations and Eq. (3.137) can further be simplified. One finally obtains −Φ ′j ( ) / λ j2

Aj

1

³

2 j

(3.138) ,

j 1, 2,3...

( ) ddr

0

After having predicted Θ1 and Θ 2 , the temperature distribution in the fluid is completely known Θ(( , )

2

ψ( )

∞

Σ

j =1

j

j

( ) exp (

(3.139)

)

With the help of Eq. (3.139), the distribution of the Nusselt number can be obtained D Nu D =

∂T ∂r

TW

(3.140) r R

Tb

=

2

ψ (1)

∞

2

Σ

j =1

j

j

( 1 ) expp (

)

For the hydrodynamically and thermally fully developed flow, the equation for the Nusselt number reduces to Nu ∞ =

2 ψ (1) + C2

(3.141)

Figure 3.17 shows the Nusselt number for the fully developed flow, for Pr = 0.71, as a function of the rotation rate N and the Reynolds number (Reich (1988), Reich and Beer (1989)). It can be seen that the Nusselt number strongly decreases with increasing rotation rate N, which is caused by flow laminarization. For N → ∞ , the Nusselt number approaches the limit Nu ∞ 4.36 for laminar pipe flow.

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux

81

Fig. 3.17: Nusselt number for fully developed flow in an axially rotating pipe (adapted from Reich and Beer (1989), Weigand and Beer (1989a))

Fig. 3.18 shows the distribution of Nu / Nu ∞ in the thermal entrance region for air flow. It can be seen that the length of the thermal entrance region is increased with increasing rotation rates. This is caused by the laminarization of the flow. For N → ∞, the thermal entrance length approaches the one for laminar pipe flow which is given by Ltth = 0.05 Re D Pr D

(3.142)

This effect has interesting implications, specially for larger Reynolds numbers. For example for a Reynolds number of about 50 000, the flow is not even thermally fully developed after 80 pipe diameters for N = 1. This shows very clearly that care has to be taken in predicting the heat transfer for such types of flows. Reich (1988) obtained experimentally the Nusselt numbers in an axially rotating pipe with constant heat flux at the wall. The rotating test section was 40 diameter in length. He compared his experimental results with calculations for a fully developed flow and found that the difference between experimental data and calculated values increased with growing values of the rotation rate. However, if one calculates the Nusselt numbers from the above solution for the thermal entrance region and compares them to the measured values, very good agreement is obtained. This is shown in Fig. 3.19 (Weigand and Beer (1989a)). The results presented here can easily be extended to other wall boundary conditions like constant wall temperature or uniform external heating or cooling. The reader is referred to Weigand and Beer (1992b) for such cases.

82

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

Fig. 3.18: Nusselt number distribution in the thermal entry length for different rotation rates N and Pr = 0.71 (Weigand and Beer (1989a))

Fig. 3.19: Nusselt numbers at x/D = 40 for various values of the rotation rate and Pr = 0.71 (Weigand and Beer (1989a))

The examples of the present chapter have shown that even complicated technical problems can be solved using analytical methods. It is important to note that in the analytical solution the physical significance of individual effects (wall bound-

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux

83

ary conditions, type of fluid, rotation,…) can be more easily identified than in the numerical solution of the problem. Problems

3-1. Consider the following eigenvalue problem Yn′′ λn2Yn = 0 with the boundary conditions Yn′(0)

n

(1)

0

a.) Prove that the eigenvalue problem is self-adjoint and positive definite. b.) Show that the eigenfunctions form an orthogonal set of functions in the interval [0,1]. c.) Calculate the eigenfunctions and the eigenvalues. Use a normalizing condition that (0) 1. 3-2. Show that the following three eigenvalue problems are self-adjoint and positive definite:

(

)λ

n

Φ n (0)

′′n

n n

(1)

0

0

Rn′′ Rn′ λn2 rR Rn = 0 rR R ′(0) (1) 0

( Yn (0)

)

′ λn (

′′ 2

n

n

(1)

n

n

1)

n

0

0

3-3. Consider the flow and heat transfer is a planar channel with height 2h. The plates have a constant wall temperature T0 for x < 0 and TW for 0 . The velocity profile of the laminar flow in the planar duct can be simplified to be a slug flow profile with u u = const. . The velocity component in the y – direction v = 0. All fluid properties can be considered to be constant. The temperature distribution in the fluid can be calculated from the energy equation u

∂T ∂xx

with the boundary conditions

a

∂ 2T ∂yy 2

84

3 Heat Transfer in Pipe and Channel Flows (Parabolic Problems)

x

0:T

y

0:

T0

∂T ∂yy

00,

y

h : T = TW

a.) Introduce the following dimensionless quantities x

T TW x a y , y= , Θ= h uh h T0 TW

What is the resulting equation and the boundary condition in dimensionless form? b.) Use the method of separation of variables to solve the given problem ( f ( ) g ( y ) ). What equations are obtained for the functions f and g? c.) Calculate the eigenfunctions and display the first and the third eigenfunction as a function of y in the interval [0,1]. d.) Predict the complete solution for the temperature field. 3-4. A planar channel with the height 2h is subjected to a hydrodynamically fully developed laminar flow. The fluid enters the channel with the constant temperature T0. The walls of the channel are adiabatic for x < 0, whereas for x ≥ 0 the walls are heated by a constant heat flux qW . Assume that all fluid properties are constant. The velocity distribution is given by 2 u 3§ y· · = ¨1− 1− ¸ ¸ u 2 © © h ¹ ¹¸

and the Peclet number is large enough, so that axial heat conduction can be ignored in the fluid. a.) Calculate the temperature distribution in the fluid for large axial values. Display the form of the radial temperature distribution for a given value of the axial coordinate. b.) Predict the value of the Nusselt number for the fully developed flow. c.) Predict the complete solution of the problem (hint: Split the problem in two separate problems). 3-5. Consider the production process of a thin foil. The foil is moved with constant velocity u∞ through a thin slit and cools down in the surrounding air. The process is schematically shown in the following figure

3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux

85

The distance between the foil and the wall is h. The wall is cooled for x > 0 and takes out heat from the surrounding air with the constant heat flux qW . For large axial values, a hydrodynamically fully developed velocity profile according to u y = 1− u h is established. The temperature distribution in the air gap should be predicted for large axial values. The upper wall (foil) can be assumed to be adiabatic and all fluid properties are constant. a.) Simplify the energy equation for the given problem. Derive the boundary conditions for y = 0 and y = h and make the energy equation dimensionless. b.) Derive an integral energy balance for the thermal fully developed flow. What functional form the temperature distribution must have for the fully developed flow? c.) Introduce the above derived expression for the temperature distribution into the energy equation and solve the equation with the given boundary conditions. d.) Calculate the Nusselt number at the lower wall. Use T ( x, y 0) T ( x, y h) as the driving temperature difference for the Nusselt number.

4 Analytical Solutions for Sturm - Liouville Systems with Large Eigenvalues

The numerical method for obtaining the eigenvalues and eigenfunctions of Eq. (3.25) is explained in Chap. 3 and Appendix C. This numerical procedure gets cumbersome and difficult if the eigenvalues become large, because of the increasing number of zero points of the eigenfunction. This is elucidated in Fig. 4.1, which shows several eigenfunctions for a turbulent pipe flow with constant wall temperature. As it can be seen from Fig. 4.1, the higher eigenfunctions strongly oscillate. It can be shown, that the eigenfunction of number j has also j zero points. This means that, for larger j, it is difficult to capture exactly the shape of the eigenfunctions by means of a numerical method. An increasingly larger number of grid points is needed to resolve the higher eigenfunctions. Because of this reason, there has been a lot of interest in obtaining analytical approximations of eigenfunctions for larger eigenvalues. Good reviews about this subject can be found in Erdelyi (1956) and in Kamke (1983). For laminar flows, the hydrodynamically fully developed velocity profile is a simple quadratic function of the spatial coordinate. In addition, the function a2 ( ) in Eq. (3.25) is equal to one. These two simplifications allow that asymptotic approximations can be obtained relatively easily for laminar flow problems. A good review on this topic is given in Shah and London (1978). In the present section, only some of this work will be presented. Sellars et al. (1956) investigated the Sturm-Liouville eigenvalue problem for large eigenvalues for pipe and channel flows with constant wall temperature. They used the WKB(J)1 method for obtaining the eigenvalues for λ → ∞ . As a result, they found that the large eigenvalues can be predicted by

λj =

4 § ¨j 2©

2· ¸ 3¹

; j 0, 1, 2, ...

(4.1)

The eigenvalues given by Eq. (4.1) are in good agreement with numerically predicted values for larger j (for j > 5 they are nearly identical). Lauwerier (1950) showed, that the solution of the laminar Graetz problem can be expressed by 1

The description WKB(J) stands for Wentzel, K Kramer, Brillonin and Jeffreys, who developed the method in the 1920s. However, the method was applied much earlier by Liouville (1837). For an introduction into the method, the reader is referred to Simmonds and Mann (1986) and Plaschko and Brod (1989). For a more detailed description of the method, the reader is referred to Froman and Froman (1965) and to Kumar (1972).

88

4 Analytical Solutions for Sturm-Liouville Systems with Large Eigenvalues

Fig. 4.1: Eigenfunctions for a turbulent pipe flow with constant wall temperature (ReeD = 10000, Pr = 0.01)

Whittaker functions (see Kamke (1983)), where the eigenvalue appears in the argument as well as in the subscript of the function. From the asymptotic behavior of these functions, Lauwerier was able to derive an asymptotic form for the eignfunctions for large eigenvalues. The value of the eigenfunction at the wall was determined to be 1/ 3

Φ(

)

3§ 6 · ¨ ¸ 3π © 2λ ¹ 1/ 3

3§ 6 · + ¨ ¸ 3π © 2λ ¹

(

ª § 2λ «¬© 4

) cos «

2p 1 º § 2 · π » ¦α p ¨ ¸ 6 ¹ ¼» p 0 © 2λ ¹

2p ª § 2λ 1 º § 2 · 3 Γ ( 2 / 3) cos « + π βp ¨ ¸ 6 ¹ ¼» p 0 © 2λ ¹ «¬© 4 1/ 3

4/3

(4.2)

4.1 Heat Transfer in Turbulent Pipe Flow with Constant Wall Temperature

α0 1 α1 0 01444444444 α 2 0 009882467268 α 3 = −0.02131664753,

89

β 0 = −0.0785714857 β1 0.02887167277 β 2 = −0.0440553529

From Eq. (4.2), the eigenvalues can be calculated for the constant wall temperature case by setting Eq. (4.2) equal to zero. Newman (1969) extended the analysis by Lauwerier and obtained the following expressions

λj

λ0

s1λ0

A j ′j ((1) 1)

4/3

λj

s2 λ0 1/ 3

8/3

s3 λ0

10 / 3

ª1 1λ0 4 / 3 « « L4 λ0 10 / 3 ¬

s 4 λ0

11 / 3

O

λ

6/3

L3λ0

λ

11/ 3

0

2 0 5 0

(

−7 / 3

( )

) º » » ¼

(4.3)

(4.4)

with the quantities

λ0 =

4 § ¨j 2©

2· ¸ 3¹

(4.5)

; j 0, 1, 2, ...

s1

0 159152288 1 91 2288

L1 = 0.144335160

s2

0 01148 0114856354 63 4

L2 = 0.115555556

s3

0 224 224731440 31440

L3 = −0.21220305

s4

0 033772601 033 2601

L4 = −0.187130142

C

2.025574576,,

L5 = −0.0918850832

(4.6)

For the case of a wall boundary condition of the third kind, Hsu (1971a) derived asymptotic expressions for the eigenvalues and related constants. He showed that his expressions are identical to those of Sellars et al. (1956) for the case of an uniform wall temperature. For the case of an insulated wall, the expressions of Hsu (1971a) are identical to those provided by Hsu (1965) for the case of a constant wall heat flux. For a turbulent internal flow, the mathematical problem of obtaining asymptotic expressions for the eigenvalues and eigenfunctions is much more complicated than for laminar internal flow. This is due to the complicated description of the velocity profile and the appearance of the eddy diffusivity in Eq. (3.25), which causes the function a2 ( ) to be a complicated function of the coordinate.

r F u(( )

2 j

j

( ) ª¬

F 2

( ) ′j ( ) º¼′

0

(3.25)

90

4 Analytical Solutions for Sturm-Liouville Systems with Large Eigenvalues

n = 0 : ′ ( 0) 0

(3.26)

(1) 0

n = 1:

Sternling and Sleicher (1962) derived asymptotic expressions for the eigenvalues and eigenfunctions for heat transfer in a turbulent pipe flow with constant wall temperature. However, it turned out that the accuracy of their results for Aj, Eq. (3.59), as well as for Aj Φ j ( ) , Eq. (3.69), was somewhat disappointing. Eight years later, Sleicher et al. (1970) were able to derive more accurate expressions using a matched asymptotic expansion. In the following sections, the method of Sleicher et al. (1970) is described in more detail for the case of a constant wall temperature, because their method is very useful for a lot of similar problems. The method has been successfully used with some modifications by Shibani and Özisik (1977a), Sadeghipour et al. (1984) and Shibani and Özisik (1977b) for different wall boundary conditions and different geometries. At the time - the method has been established by Sleicher et al. (1970) - the computational power of the computers did not permit a detailed comparison between their derived asymptotic expressions and numerical solutions of the eigenvalue problem for larger eigenvalues. Therefore, a short section has been added at the end of this chapter which shows a detailed evaluation of the asymptotic expressions compared to numerical calculations. In addition, it is shown, that the method of Sleicher et al. (1970) can be adapted quite easily to other related problems. This is demonstrated for the heat transfer process in an axially rotating pipe. We start our considerations with the Sturm-Liouville problem given by the Eqs. (3.25-3.26) and consider a turbulent pipe flow. For this type of application, the equations read ((F = 1, n r ) dΦ j º ª « r a2 ( ) » ddr ¼ ¬

d dr

( )λ j

(4.7)

2 j

( )

0

with the boundary conditions

r = 0 : ′ ( 0) 0 r = 1:

(1)

(4.8)

0

In order to obtain an asymptotic expression for large eigenvalues, it is common to transform the Sturm-Liouville system into its standard form. This can be done by introducing the following new quantities (see Kamke (1983)) G

1

π

u ddrr,, k 2 2

1

³ 0

1/ 4

ϑ

§ r 2 a2 u · ¨ ¸ © 2 ¹

j,

w(

2 G, ξ

1

r

³ 0

§ r 2 a2 u · ¸ © 2 ¹

)=¨

−1/ 4

u dr 2 a2

(4.9)

1/ 4

d 2 § r 2 a2 u · ¨ ¸ dξ 2 © 2 ¹

(4.10)

4.1 Heat Transfer in Turbulent Pipe Flow with Constant Wall Temperature

91

One obtains d 2ϑ + ªk 2 dξ 2 ¬

w(

(4.11)

) º¼ ϑ = 0

with the boundary conditions

0 : ϑ ( 0) 0

ξ

ξ π :ϑ(

)

(4.12)

0

After introducing the new coordinate ξ, the interval for the coordinate has changed from [0,1] to [0,π]. Before trying to find a solution for the eigenfunction ϑ for larger eigenvalues, one has to analyze the function w ( ) in greater detail: the function w ( ) has singularities at both boundaries ( ) . However, the function can be approximated near both boundaries with sufficient accuracy. Sternling and Sleicher (1962) subdivided the interval into two regions (core region and near wall region). For both regions, they determined solutions for Eq. (4.11) and patched them together. This procedure resulted, however, in a disappointing accuracy for the constants Aj. Therefore, Sleicher et al. (1970) choose a different approach. The interval for ξ [0,π] is split up in three separate regions (core region, near wall region and middle region). The solutions for the separate regions have to be connected later to each other by expansions of the variables2. In order to find a solution for Eqs. (4.11-4.12) for large eigenvalues, one can use the following expansion (see van Dyke (1962) or Kevorkian and Cole (1981))

ϑ

(4.13)

N

¦δ ( ) ( ) i

i =0

The functions δ i ( ) are a priori not known and have to be determined during the analysis. In order to show more clearly the solution procedure of Sleicher et al. (1970), these functions are introduced here from the very beginning. Later, during the calculation procedure, we clarify why exactly this sequence of functions is chosen for the present example

δ0 δ4

k k

1/ 2

δ1

5/ 2

δ5

3/ 2 5/ 2

δ2

l ln

(

)

3/ 2

δ3

5/ 2

ln k

(4.14)

2

Introducing Eq. (4.14) into Eq. (4.13) one obtains the following expression for the eigenfunctions ϑ k 1/ 2ϑ0 k 3 / 2 ln l k ϑ1 k 3 / 2ϑ2 + (4.15) 2 + k 5 / 2 ln l k ϑ3 k 5 / 2 4 k 5 / 2 ( ) ϑ5 2

The analysis presented here follows the excellent work of Sleicher et al. (1970). It should serve as an example on how to solve such an difficult eigenvalue propblem.

92

4 Analytical Solutions for Sturm-Liouville Systems with Large Eigenvalues

After describing the asymptotic expansion coefficients for ϑ, approximations for the function w ( ) have to be derived. Because of the singularities of the function near the boundaries, the behavior of w (

)

has to be examined separately for all

three regions. Pipe Center Region

Near the center of the pipe, the axial velocity component can be approximated with good accuracy by a constant value U U. The distribution for a2 can be approximated for this area by a constant A. If one introduces this assumptions into Eqs. (4.9-4.10) one obtains:

ξ = π r w(

) ξ

1/ 2

(4.16)

d2 1/ 2 ξ 2 ( ) dξ

1 −2 ξ 4

(4.17)

Middle Region

In this area, the function w (

) has

no singularities. Since we are interested in

solutions for large eigenvalues, the function w can be neglected compared to k2 in Eq. (4.11).

Near Wall Region

At the boundary ( ξ = π ), the function w (

)

has a second singularity. Very close

to the wall, the velocity can be approximated by u

yW

(

) . The function a2

tends to one in this region. If we introduce the above assumptions into Eq. (4.10), we obtain for w ( ) after neglecting quadratic terms w(

)

β1

(

)

2

, β1 =

5 36

(4.18)

The above expression for w ( ) was first derived by Sternling and Sleicher (1962)3. Clearly, the region, where Eq. (4.18) is fully valid, is very small. There3

The constant β1 is determined by the velocity distribution near the wall. The value of the constant is obtained under the assumption of a linear velocity distribution of the form

u

2V yW ,

V=

Re D c f 1 § du · = ¨ ¸ 2 © dyW ¹ y = 0 32 W

4.1 Heat Transfer in Turbulent Pipe Flow with Constant Wall Temperature

93

fore, Sleicher et al. (1970) used the following expression for approximating the function w ( ) in the near wall region w(

)≈−

β1

(

)

2

+

β2

(4.19)

π −ξ

Equation (4.19) assumes that the function w (

) can be expanded into a Laurent se-

ries around ξ = π . Equation (4.19) contains the not yet determined constant β2. This constant is determined during the solution process. After having investigated the functional form of w ( ) for the different regions, the determination of the eigenfunctions from Eq. (4.11) for the three different regions can be started. Pipe Center Region

In the pipe center, the function w ( sion into Eq. (4.11) results in

)

is given by Eq. (4.17). Inserting this expres-

d 2ϑC ª 1 º + 1 ϑC ds 2 ¬ 4 s 2 ¼»

0

(4.20)

where the stretched coordinate s = kξ

(4.21)

is used. The new coordinate s has the advantage that the eigenvalue does not appear explicitly in the differential equation. Figure 4.2 shows the different coordinates and the geometry under consideration. Introducing the expansion defined in Eq. (4.15) into Eq. (4.20) results after collecting terms of the same order in k in the following set of equations d 2ϑiiC ª 1 º + 1 1+ + 2 » ϑiiC = 0, 2 ds ¬ 4s ¼

0,1,...,5

(4.22)

where c f is a friction factor (Sleicher et al. (1970)). If one uses a velocity distribution according to

u = 2c (1 − r )

1/ m

one obtains for the constant β1

β1 =

, c=

1§3 1 · + 2¸ ¨ 2 4© m ¹

1 1 4m 2 4( )

The reader is referred to Shibani and Özisik (1977a) for this approach.

94

4 Analytical Solutions for Sturm-Liouville Systems with Large Eigenvalues

Fig. 4.2: Geometry and different coordinate systems used

Equation (4.22) can be transformed into a Bessel equation after introducing the new function s ϑiiC

ϑiC

(4.23)

The solution of this Bessel equation, which is finite for s = 0, is given by

ϑiiC = Bi s J 0 ( ) ,

(4.24)

0,1,...,5

The complete solution for the pipe center region is therefore given by

ϑC = s J 0 (

­ B0 k ) °® °¯ B3 k

1/ 2 5/ 2

B1 k

3/ 2

l k B2 k −3 / 2 ln

lln k B4 k

5/ 2

B5 k

5/ 2

(

)

2

½° ¾ °¿

(4.25)

From the definition of the function ϑ in Eq. (4.10) it follows that in the pipe center ( r → 00, 2 ,  , j ( 0 ) 1 ), the function ϑ will behave like 1/ 4

ξ → 0 :ϑC =

§ r 2 a2 u · © 2 ¹

Φj (

)=

GA Aξ

(4.26)

where the normalizing condition for the eigenfunction Φ j ( ) 1 has been used. If one introduces now the stretched coordinate s into Eq. (4.26), one obtains in the center of the pipe ξ 0 :ϑC ( ) GAξ , s 0 : k 1/ 2 GA s (4.27) This expression can be compared to Eq. (4.25) for s → 0 (J0 (0) = 1). This results in

4.1 Heat Transfer in Turbulent Pipe Flow with Constant Wall Temperature

B0 + B1 k 1 ln k B2 k

1

B3 k 2 ln k B4 k

2

B5 k

2

(

k)

2

GA

95

(4.28)

Because Eq. (4.28) has to be valid for all k, it follows B0 Bi

(4.29)

GA 0, 0

1,...,5

And the solution for the center region of the pipe is

ϑC = k

1/ 2

GA s J 0 (

)

(4.30)

Middle Region

In the middle region, the function w ( ties. Because of this, w (

)

) is a continuous function without singulari-

can be neglected in Eq. (4.11) compared to the large

eigenvalues. After introducing again the stretched coordinate s, one obtains from Eq. (4.11) d 2ϑM + ϑM = 0 ds 2

(4.31)

Introducing the expansion according to Eq. (4.15) into Eq. (4.31) results after collecting terms of the same order in k, in the following set of equations d 2ϑiiM + ϑiiM = 0, ds 2

(4.32)

0,1,...,5

The solution of Eq.(4.32) can be calculated easily and is given by

ϑM = k k k

( 3/ 2 ( 5/ 2 ( 1/ 2

) ) )

( (

3/ 2 5/ 2 5/ 2

)+ )+

( ln ) ( 2

(4.33)

)

The constants Ki and Li, which appear in Eq. (4.33), are yet not known. In contrast to the constants in the solution for the pipe center, they have to be determined during the following solution process. Near Wall Region

For the near wall region, the function w ( expression into Eq. (4.11) results in

)

is given by Eq. (4.19). Inserting this

96

4 Analytical Solutions for Sturm-Liouville Systems with Large Eigenvalues

d 2ϑW ª β1 + 1 2 dt 2 ¬ t

β2 º

ϑW kt ¼»

(4.34)

0

where the new stretched coordinate k(

t

)

kπ

(4.35)

s

is introduced. Inserting the expansion according to Eq. (4.15) into Eq. (4.35) results, after collecting terms of the same order in k, in the following set of equations d 2ϑiiW ª β1 º + 1 2 » ϑiW dt 2 ¬ t ¼

0 0,

(4.36)

0,1

d 2ϑ2W ª β1 º + 1 + 2 » ϑ2 dt 2 ¬ t ¼

β2

d 2ϑ3W ª β1 º + 1 + 2 » ϑ3 dt 2 ¬ t ¼

β2

d 2ϑ4W ª β1 º + 1 + 2 » ϑ4W dt 2 ¬ t ¼

β2

ϑ0W

(4.37)

ϑ1W

(4.38)

ϑ2W

(4.39)

0

(4.40)

t

t

t

d 2ϑ5W ª β1 º + 1 2 » ϑ5W dt 2 ¬ t ¼

The solution of the homogenous differential equations (4.36) and (4.40) can be obtained in the same way as for Eq. (4.22). Introducing again the new function t ϑiiW

ϑiW

(4.41)

into Eqs. (4.36) and (4.40) results in

t 2ϑiW

t

iW

(

(β

) )ϑiW

(4.42)

0

As it can be seen, Eq. (4.42) is a Bessel equation (see Bowman (1958)). For β1 = 5 / 36 4 (this is the value used by Sternling and Sleicher (1962) for the linear velocity distribution near the wall) one obtains t 2ϑiW

t

iW

(

)ϑ

iiW

(4.43)

0

This equation has the general solution

ϑiiW = t {Di J1/ 3 ( 4

)

J

( )} ,

0,1,5

(4.44)

For other values of β1, the reader can find some solutions in Shibani and Özisik (1977a).

4.1 Heat Transfer in Turbulent Pipe Flow with Constant Wall Temperature

97

Equation (4.44) is also the solution of the homogenous Eqs. (4.37)-(4.39). A particular solution for these equations can be constructed by using the method of the variation of constants (see Bronstein and Semendjajew (1981)). This leads finally to the complete solution for the near wall region, given by ϑW = k

1 / 2

+ k

−k

³

ª « « « « « « « « « ¬

³

−5/2

πβ

J -1 / 3 J −5/2

t

2

­ ° ® ° ¯

J1/3 J1/3

+k

3

­ ° ® ° ¯

J1/3

π

β

3

5 / 2

β

3

t

§ ¨ ¨ ©

2

β

2

J1/3 J1/3 E 0 J 1 / 3 J -1 / 3 0

· ¸ ¸ ¹

D

2

½ ° ¾ + °¿

E

2

½ ° ¾ °¿

dt

D 1 J 1 / 3 J -1 / 3 E 1 J -1 / 3 J -1 / 3

· ¸ ¸ ¹

§ D1 J1/3 J1/3 ¨ ¨ E 1 J -1 / 3 J 1 / 3 ©

dt

D

+

3

½ ° ¾ °¿

½ E 3 °¾

· ¸ dt ¸ ¹

°¿

D

· J 1 / 3 J -1 / 3 ¸ d t E 0 J - 1 / 3 J - 1 / 3 ¸¸ 0

¹

§ ¨ 2 ¨ ¨ ©

D

· J1/3 J1/3 ¸ d t E 0 J - 1 / 3 J 1 / 3 ¸¸ 0

¹

D

½ ° 2 ¾d t ° ¿ ½ E 2 °¾ d ° ¿

t

D

º » » » » » » » 4 » » ¼

J -1 / 3
100. These results are in good agreement with the investigation of Acrivos (1980), who showed analytically that the region of influence for axial heat conduction, scaled by the pipe radius, is of the order of Pe −D1 . Nguyen (1992) and Bilir (1992) investigated numerically the heat transfer for thermally developing flow in a circular pipe and in a planar channel. Nguyen (1992) derived from his computational results accurate engineering correlations for the Peclet number effect on the local Nusselt number in the thermal entry region. Although axial heat conduction can be ignored for turbulent convection in ordinary fluids and gases, with liquid metals this may not always be justified. In fact, due to the very low Prandtl numbers for liquid metals, the Peclet number can be as small as three in turbulent duct flows. Reviews on the convective heat transfer in liquid metals can be found in Reed (1987), Stein (1966) and Kays and Crawford (1993). Lee (1982) studied the extended Graetz problem in turbulent pipe flow. He found that, for Peclet numbers smaller than 100, axial heat conduction in the fluid becomes important in the thermal entrance region. He investigated a pipe which was insulated for x ≤ 0 and had a uniform wall temperature for x > 0. Lee (1982) used the method of Hsu (1971b) to obtain a series solution for the problem. In this work, the variation of the Nusselt number for fully-developed flow was shown for several Peclet and Prandtl numbers. In addition, the error in Nusselt number resulting from neglecting the axial heat conduction effect was presented. This error was found to be as high as 40% for a Peclet number of 5. For the case of turbulent flow inside a parallel plate channel, the effect of axial heat conduction within the fluid was studied by Faggiani and Gori (1980). They solved numerically the energy equation for a constant heat flux boundary condition. Weigand (1996) extended the method of Papoutsakis et al. (1980a) to solve the extended Graetz problem for turbulent flow in a pipe and in a planar channel. Like the solution given by Papoutsakis et al. (1980a) for laminar internal flow, the solution by Weigand (1996) has not been plagued by the uncertainties arising from expansion in terms of eigenfunctions, which belong to a non self-adjoint eigenvalue system. Later, Weigand et al. (1997) and Weigand and Wrona (2003) extended the method also for heat transfer in concentric annuli. Nearly all studies mentioned here considered only a heating section semiinfinite in size. Hennecke (1968) was one of the few who calculated numerically the temperature distribution at the end of the heating section for laminar internal flow. However, he investigated this problem only for a very large heating section. The problem was solved later analytically by Papoutsakis et al. (1980b) for laminar pipe flow and a heating zone of finite length. They derived expressions for the Nusselt number and the bulk-temperature, but did not show, however, any distributions for these quantities. Only temperature profiles in the fluid were shown.

120

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

Weigand et al. (2001) calculated the effect of a changing length of the heating zone for laminar and turbulent internal flow and for different Peclet numbers. One of the main results of their study was that, for finite heated sections, the axial heat conduction effect depends also on the length of the heated zone. Axial heat conduction may, therefore, be important also for Peclet numbers larger than 100 if this zone is short. The present chapter shows an analytical solution method for the extended Graetz problem. The method is based on the approach by Papoutsakis (1980a, 1980b) and can be used for a lot of related problems. First, the method is explained for a duct with a semi-infinite heating section (with constant wall temperature or constant wall heat flux) to highlight the basic ideas behind the solution approach. Later applications of the method to piecewise heated ducts are explained. The goal of this chapter is to show that the derived final solutions for the extended Graetz problem are as simple as the ones for the parabolic problems discussed in Chap. 3.

5.1 Heat Transfer for Constant Wall Temperatures for x≤ ≤0 and x > 0 We start our considerations for a hydrodynamically fully developed flow with a low Peclet number and constant wall temperature. As for the analysis in Chap. 3, it is assumed that the velocity profile is fully developed. This means that the vertical velocity component v is zero and the axial velocity component is only a function of the coordinate orthogonal to the main flow direction. The geometry under consideration and the used coordinate system is shown in Fig. 5.1.

Fig. 5.1: Geometry and coordinate system (Weigand (1996))

If we assume constant fluid properties, the energy equation is given by Eq. (3.5) and Eq. (3.6)

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

121

Pipe

ρc p u ( r )

∂T ∂x

1∂ª ∂T r k r ∂r ¬ © ∂r

ρ cp v ' T '

∂ ª T k y¬ y

c p v 'T '

º ∂ ª ∂T º k ρ c p u ' T '» ¹¼ ∂x ¬ ∂x ¼

(3.5)

Planar Channel

ρcp u ( y )

∂T ∂xx

º

∂ ª ∂T k x x

¼

º c p u ' T '» ¼

(3.6)

If one analyses Eq. (3.5) or (3.6) for laminar flow ( u' T ' 0 , v' T ' 0 ), one finds immediately that the equations are elliptic in nature. This is due to the axial heat conduction effect within the flow, which is represented by the second term on the right hand side of the equations. For turbulent flow, the nature of the equation will also depend on the turbulent heat fluxes. The turbulent heat fluxes − ρ cP v ′T ′ and − ρ cP u ′T ′ have to be modeled. This can be done for example by using a simple eddy viscosity model1: −v' T ' = ε hy

∂T (planar channel), ∂yy

−u' T ' = ε hx

−v' T ' = ε hr

∂T (pipe) ∂r

∂T (planar channel and pipe) ∂xx

(3.7)

(3.8)

where ε hy , ε hr and ε hx are only functions of the coordinate orthogonal to the flow direction. Inserting Eqs. (3.7-3.8) into Eq. (3.5) and Eq. (3.6) results in Pipe

ρcp u ( r )

1

∂T ∂x

1 ∂ ª r( r ∂r «¬

) ∂∂Tr º»

∂ ª «( ¼ ∂x ¬

) ∂∂Tx º» ¼

(3.9)

More advanced models can also be used, which assume the stream-wise component of the turbulent heat flux to be also a function of the cross-stream temperature gradient and vice versa (see Batchelor (1949), So and Sommer (1996) and Sommer (1994)). Weigand et al. (2002) conducted a numerical analysis of the energy equation (3.6) and found the above mentioned simple model to be in perfect agreement with more complicated models for the range of parameters considered here.

122

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

Planar Channel

ρcp u ( y )

∂T ∂xx

∂ ª ( y¬

Tº ∂ ª + ( y »¼ x «

)

) ∂Tx º» ¼

(3.10)

If one replaces in Eqs. (3.9-3.10) the heat diffusivities ε hy , ε hr and ε hx by a turbulent Prandtl number according to Eq. (3.18), one obtains Pipe

ρcp u (r )

∂T ∂x

1 ∂ ª r k r ∂r ¬ ©

cp

Tº Prt ¹ r ¼

ª

εm

x ©

k

cp

ε hhx · ∂T º » Prt ε hr ¹ ∂xx ¼ P m

(5.1)

Planar Channel

ρcp u ( y )

∂T ∂xx

∂ ª k y ¬©

cp

Tº Prt ¹ y ¼ m

∂ ª k x «¬©

cp

m

hhx

Pt Pr

hy hy

∂T º » ¹ x »¼

(5.2)

Eqs. (5.1-5.2) can be combined to one equation if we introduce, like in Chap. 3, a vertical coordinate n which is equal to y for a planar channel and equal to r for pipe flows. This results in

ρcp u ( n)

∂T ∂x

1 ∂ ª F r k F ∂n ¬ ©

cp

Tº Prt ¹ n ¼

εm

ª x ¬©

k

cp

· ∂T º » Prt ¹ ∂xx ¼ P m

(5.3)

where the superscript F specifies the geometry and has to be set to zero for a planar channel and equal to one for pipe flow. The boundary conditions for Eq. (5.3) are n L:

,

0

T TW ,

0

0

(5.4)

n =0 : ∂ / ∂ =0 lim lim T TW 0 , x → −∞

x → +∞

Introducing the following dimensionless quantities into the above equations Θ=

μ cp T TW r x 1 n uL , r = , x = , n = , Pr = , ReL = , T0 TW R L ReL Pr L k ν εm =

εm u , u = , P eL ν u

ReL Pr

(5.5)

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

123

where the length scale L = R for a pipe flow and L = h for the flow in a planar channel, one obtains u (

)

∂Θ 1 ∂ ª = 2 a1 ( ∂x P Pee L ∂x «¬

)

∂Θº 1 ∂ ª F + r a2 ( ∂ x »¼ r F ∂n «¬

)

∂Θº ∂n »¼

(5.6)

with the boundary conditions n = 11: Θ =11,  ≤ 0 Θ = 0,  > 0 n = 0 : ∂Θ / ∂ = 0 lim 0, lim x → −∞

The functions a1 (

)

and a2 (

a1 ( ) 1

x → +∞

)

Pr m , Prrt

(5.7)

1

are given by

2

( ) 1+

· Pr § ε hx h ¨ ¸ εm Prt © ε hn ¹ Pr

(5.8)

Papoutsakis et al. (1980a) showed that it is possible to solve Eq. (5.6) for laminar pipe flow ( a1 a2 11, 1 ) by decomposing the elliptic partial differential equation into a pair of first order partial differential equations. The ensuing procedure for solving the extended turbulent Graetz problem given by Eqs. (5.6-5.7) follows the method of Papoutsakis et al. (1980a) for deriving the solution of the more general problem where a1 and a2 are functions of n . In order to obtain a solution for the turbulent Graetz problem given by Eqs. (5.6-5.7), one can proceed in the following way: First, we define a function E ( , ) which may be called the axial energy flow through a cross - sectional area of height n . This function is given by n ª º F 1 E = ³ «u Θ − 2 a1 (n ) d » r dn Pe L P ∂xx ¼ 0 ¬

(5.9)

If one introduces the axial energy flow according to Eq. (5.9) into the energy equation (5.6), one obtains the following system of first order partial differential equations → ∂ → S( , ) ( , )  ∂xx G where the solution vector S and the matrix operator L are defined by 

(5.10)

124

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

ª Pe L u ( ) « a ( ) L =« 1  « F ∂ « r a2 ( ) ∂n ¬

G ªΘ ( , ) º S « », ¬ E(( , ) ¼

−

Pe 2L ∂º » F r a1 ( ) ∂n » » 0 » ¼

(5.11)

The boundary conditions for the function E can be obtained directly from its definition, Eq. (5.9), by using Eq. (5.7). This results in n

lim E

x → −∞

³u r

F

ddn ,

0

lim

x → +∞

0

0:

(5.12)

0

Before solving Eq. (5.10), some interesting features of the operator L and the cor responding eigenvalue problem for this equation should be presented. The most remarkable aspect of L is that it gives rise to a self-adjoint problem even though  the original convective diffusion operator is non self-adjoint. This fact is of course dependent on the sort of inner product between two vectors, which will be used. This means that we have to find an inner product which is symmetric. Papoutsakis et al. (1980a) obtained such an inner product for a laminar pipe flow. It is now possible to extend this work for laminar and turbulent duct flows. Let us define the following inner product for two vectors G ªΦ ( Φ=« 1 ¬Φ2 (

)º ) »¼

ªϒ ( ,ϒ=« 1 ¬ 2(

)º ) »¼

1 a2 ( n ) r F

2

(5.13)

by G G Φ,

1

³ 0

ª a1 ( n) r F « PeL2 ¬ P

1

(n))

1

( n)

( n)

2

º (n) » ddn ¼

(5.14)

G G The inner product of the two vectors , ϒ incorporates the two functions a1 a2 which contain the heat diffusivity for the turbulent flow. In case of a laminar pipe flow, the two functions will be equal to one. L belongs to the following domain  G G (5.15) D(( ) : (exists and) , 1 (1) 2 (0) 0   Then it can be shown that the matrix operator L is a symmetric operator in the  Hilbert space H H. This means that G G G G (5.16) Φ , L ϒ = LΦ, ϒ   G G Eq. (5.16) can be proven by inserting the expressions LΦ and Lϒ into the defini  tion of the inner product according to Eq. (5.14). This results in

{

}

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

G G Φ,  G G LΦ , 

1

³ 0

1

³ 0

ªu r F « PeL ¬P

1

(n )

1

( n)

1

( n)

2

( n)

2

( n)

1

º ( n) » d dn ¼

ªu r F « PeL ¬P

1

(n )

1

(n)

2

( n)

1

( n)

1

( n)

2

º ( n) » d dn ¼

Substracting Eq. (5.18) from Eq. (5.17) results after integration in G G G G Φ , ϒ − Φ, ϒ = Φ 2 ( ) ( ) ( ) ( )   − Φ1 ( ) ( ) ( ) ( )

125

(5.17)

(5.18)

(5.19)

The resulting expression on the right hand side of Eq. (5.19) is zero, if the vectors G G Φ and ϒ satisfy the conditions given in Eq. (5.15). The associated eigenvalue problem to Eq. (5.10) is given by G G (5.20) LΦ j = λ j Φ j  G In this equation Φ j denotes the eigenfunction corresponding to the eigenvalue

λ j . If we introduce the definition of the matrix operator L into this equation, the 

following eigenvalue problem is obtained

ª u Pe 2L « Φ j1 − ¬ a1 ( )

º Φ ′j 2 = λ j Φ j1 1( ) ¼

1

r F a2 ( ) Φ ′j1 = λ j Φ j 2

(5.21)

(5.22)

From the above system of ordinary differential equations, one differential equation for the first component of the eigenvector Φ j1 can be easily obtained by eliminating Φ j 2 . This results in

ª¬ r F a2 (n)

j1

º¼′ + r F

ª λ j a1 ( ) º − u λ j Φ j1 = 0 Pe 2L ¬ P ¼

(5.23)

Eq. (5.23) has to be solved together with the homogenous boundary conditions Φ ′j1 (0) 0

,

j1

(1) 0

(5.24)

Usually Eq. (5.23) is solved numerically with the procedure described in Appendix C. The method used is a “shooting method”, where the differential equation (5.23) is integrated from the centreline of the duct ( 0 ) to the wall and then the boundary condition at the wall Φ j1 ( ) 0 is checked for an assumed value of λ j .

126

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

Therefore, an additional condition for that Φ j1 (

0 is required. A possible condition is

)

1

(5.25)

This condition is a normalizing condition for the eigenfunctions Φ j1 . Eq. (5.23) shows clearly that the eigenvalues appear as λ j and λ j2 in the equation. This is different to the normal Graetz problem (see Eq. (3.25)). For the eigenvalue problem according to Eq. (5.20) it is possible to show that it is self-adjoint and semidefinite (see Sauer and Szabo (1969)). Therefore, all eigenvalues of Eq. (5.23) have to be real. The real eigenvalues and the associated eigenfunctions of Eq. (5.23) define a complete set of orthogonal functions (see Appendix D). With the help of these eigenfunctions any function can be reconstructed. Because the matrix operator L is neither positive nor negative definite, one will find both positive  eigenvalues and negative ( ) . The eigenfunctions belonging to these eigenvalues, ( ) , form an orthogonal basis in the Hilbert space H. It is interesting to note that Eq. (5.23) reduces to the eigenvalue problem for the parabolic case, Eq. (3.25), if the Peclet number tends to infinity. Because the two sets of eigenvectors, normalized according o to equation (5.25), constitute an orthogonal basis in → H, an arbitrary vector f can be expanded in terms of eigenfunctions in the following way ∞

→

→

f = ¦ Dj Φ j ( )

(5.26)

j=0

The constants Dj in this equation can be calculated if we apply to both sides of Eq. (5.26) the inner product according to Eq. (5.14). This results in G G f, i

∞

¦D

j

G G Φ j , Φi

(5.27)

j =0

G G Because the expression Φ j , Φ i

is equal to zero for i ≠ j (see Appendix D), Eq.

(5.27) can be resolved for the constants Dj and one obtains G G G G f, j f ,Φ j Dj = G G = G 2 Φ j ,Φ j Φj

G where the vector norm Φ j

2

(5.28)

has been introduced into Eq. (5.28). Inserting Eq.

(5.28) into Eq. (5.27) results in the following expression for the expansion of an G arbitrary vector f in terms of eigenfunctions

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

→

∞

f =¦ j =0

G G f ,Φ j → G 2 Φj( ) Φj

127

(5.29)

If we explicitly distinguish in equation (5.29) between positive and negative eigenvectors, the equation takes the following form G G G G (5.30) ∞ f , +j f , Φ −j G − → + f =¦ Φj ( ) + Φj ( ) 2 2 j 0 j 0 Φ +j Φ −j After the expansion of an arbitrary vector in terms of eigenfunctions has been developed, we can return to the solution of the system of partial differential equations given by Eq. (5.10). Because of the non-homogeneous boundary condition G for the temperature, the vector S does not belong to the domain D ( ) given by Eq. (5.15). However, it can be shown that G G G G L S, j S, L j  holds (see Appendix D). The function g (

j2

)

(1) g ( x)

(5.31)

is for the case of a semi-infinite heat-

ing zone given by  0  0

­1, g ( x) = ® ¯0,

(5.32)

If we apply the inner product, given by Eq. (5.14), to the system of partial differential equations, Eq. (5.10), one obtains ∂ G G S, ∂xx

G G LS , Φ j 

j

(5.33)

Inserting Eq. (5.31) into Eq. (5.33) (and taking Eq. (5.20) into account) results in ∂ G G S, ∂xx

j

λj

G G ,

j

( )

j2

(1)

(5.34)

G G Eq. (5.34) represents an ordinary differential equation for the expression S , Φ j . For the homogenous equation one obtains the solution G G S, j e p(λ j ) 0 j exp(

(5.35)

A particular solution of Eq. (5.34) can be found by using the method of the variation of constants (see e.g. Bronstein and Semendjajew (1983)). One obtains after splitting the solution into positive and negative eigenvalues

128

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

G G S,

j

0j

x

³(

exp((λ j )

j2

((1)) )) exp((λ j ( x

x )) ddx

+ j2

((1)) )) eexp( p(( λ j+ ( x

x )) ddx

( )

(5.36)

−∞

G G S , +j

∞

³ ( g ( x)

C0+ j eexp( p((λ j+ x )

(5.37)

x

Because the solution must be bounded for x → + ∞ and for x → − ∞ , the two constants C0−j and C0+j , appearing in Eqs. (5.36-5.37) have to be zero. The integral in Eqs. (5.36-5.37) can be evaluated by inserting the function g (

)

into these equa-

tions. One obtains x

G G Φ +j 2 ( ) 0 : S,Φ j = expp (

λj

x

(1)

+ j2

)−

λ

+ j

G G Φ −j 2 ( ) 0 : S,Φ j = exp (

−

j2

λ

()

(5.39)

)

λj

(5.38)

− j

G Introducing these expressions into the expansion for the function S , Eq. (5.29), G and taking the first vector component of S , the following result is obtained for the temperature distribution in the fluid ∞

x ≤ 0 : Θ ( , ) = − ¦

Φ −j 2 (1)

λj

∞

Φ +j 2 (1)

j =0

∞

j 0

+ j1

( ) 2

Φ −j 2 (1)

j =0

+ j2

−¦

j

G λ j+ Φ +j

x > 0 : Θ( , ) = − ¦

( )

2

j 0

¦

j1

(1)

+ j1

λ j+ Φ

+ 2 j

( )

+

(5.40)

exp(λ j+ )

G λ −j Φ −j

j1

( )

2

exp(λ j )

(5.41)

From Eqs (5.40–5.41) it can be seen that the solution for x ≤ 0 contains both, negative and positive eigenfunctions, whereas for x > 0 only the negative eigenfunctions are needed for the solution of the problem. For Eq. (5.40), it is possible to show that ∞

−¦

j=0

Φ −j 2 (1)

G λ −j Φ −j

( )

j1 2

∞

−¦ j =0

Φ +j 2 (1)

G λ j+ Φ

+ j1 2 + j

( )

=1

(5.42)

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

129

This can be shown by expanding the vector (1,0)T into a series according to Eq. (5.29) (see Appendix D for a derivation of this result). Introducing this result into Eq. (5.40) leads to ∞

x ≤ 0 : Θ ( , ) =1 + ¦

Φ +j 2 (1)

+ j1

G λ +j Φ +j

j =0

( ) 2

(5.43)

exp(λ +j )

From Eqs. (5.41) and (5.43) one can see that they result in a continuous temperature distribution for x = 0. In addition, one notices from the Eqs. (5.41) and (5.43) that these equations satisfy all the boundary conditions given by Eq. (5.7). If the Peclet number increases, the eigenvalues λ j+ tend to infinity and Eq. (5.43) (the temperature distribution for Θ(

0 ) results in

) = 1,

for Pe L → ∞

(5.44)

which represents the solution for the parabolic case. Eq. (5.44) also elucidates, that it is only correct to prescribe a temperature boundary condition for 0 for the limiting case of Pe L → ∞ . G 2 The vector norm Φ j which has been used in the above given solution for the temperature field, can be rewritten by using the Eqs. (5.21-5.22). One obtains (see Appendix D) G Φj

2

=−

1

1

λj

F ³r u

1

2 j1

dn

0

2 r a1 ( n ) Pe2L ³0

2 j1

dn d

(5.45)

Eq. (5.45) shows an interesting result: the second term in this equation depends on the Peclet number and tends to zero for increasing values of PeeL. This means that, for large Peclet numbers, the second term in Eq. (5.45) can be neglected and therefore, the temperature distribution, given by Eq. (5.41), is formally identical to the distribution for the parabolic problem. In addition, it can be shown from the eigenvalue problem, Eq. (5.23), that this equation reduces to one for the parabolic problem for Pe L → ∞ . This shows nicely that the results presented here approach those given in Chap. 3 for large Peclet numbers. Furthermore, it can be shown that

G Φj

2

= Φ j 2 (1)

d Φ j1 (1) dλ

(5.46) λ

j

(see Appendix D). If one introduces the abbreviation § ddΦ (1) j1 Aj = ¨ λ j ¨ dλ λ ©

λj

· ¸ ¸ ¹

−1

(5.47)

into the equation above, one finally obtains for the temperature field in the flow

130

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

∞

x 00:

¦

( , ) 1

+ j

+ j1

( ) exp( e p((λ j+ )

(5.48)

( ) exp( e p((λ j )

(5.49)

jj11

j =0

∞

¦A

x 00: Θ ( x , n )

j

j=0

After the temperature field in the fluid is known, the bulk-temperature and the Nusselt number can be calculated. Introducing Eqs. (5.48-5.49) into the definition of the bulk-temperature, Eq. (3.67), and of the Nusselt number, Eq. (3.66), results in x 00: Θb 1 2F

­° ′jj11+ ((1)) λ j+ 1 a1 ( ) ® + P 2L 0 Pe °¯ λ j

( )

F

j1 1

­° ′j1j1− ((1)) λ j− 1 A a1 ( ) ¦ j ® − P 2L 0 Pe j=0 °¯ λ j

( )

F

j1 1

∞

¦A

j

j=0

x 00: Θb 1 2 F

∞

∞

−4¦

½° e p(λ j+ ) ¾ exp( °¿ ½° e p(λ j ) ¾ exp( °¿

′j1+ (1) ( ) exp( e p((λ j+ )

+ j

(5.50)

(5.51)

(5.52)

j =0

x ≤ 0 : Nu D = 4F

­° ′jj11+ (1) ()

∞

¦

j

j =0

°¯ λ

λ +j Pe P

j

1

2 L 0

∞

−4¦

1

( n)

j1

½° (n) r F ddn ¾ exp( e p((λ j+ x) °¿

′j1 (1) ( ) exp( e p((λ j )

j

(5.53)

j =1

x > 0 : Nu D = 4F

­° ′jj11− ((1))

∞

¦

j

j =1

°¯ λ

λ j− Pe P

j

1

2 L 0

1

( n)

j1

½° (n) r F ddn ¾ exp( e p((λ j x) °¿

where F = 0 denotes the heat transfer in a parallel plate channel, whereas F = 1 denotes the heat transfer in a pipe. From Eqs. (5.52-5.53) the Nusselt number for the fully developed flow can be obtained. Investigating the limiting case x → ∞ in Eq. (5.53), only the first term of the sum needs to be retained and one obtains Nu ∞

4 λ0 4F

­° / ®1 + °¯

λ0− 2 01

(1) P (1) PeL2

1

a1 (n) 0

01

½° (n) r F dn d ¾ °¿

(5.54)

This equation shows again, that for Pe L → ∞ the Nusselt number for the fully developed flow is given by an expression similar to Eq. (3.70). For finite values of the Peclet number, this value is changed by the second term in the denominator of Eq. (5.54).

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

131

5.1.1 Heat Transfer in Laminar Pipe and Channel Flows for Small Peclet Numbers

For laminar flows the functions a1 and a2 are equal to one. In addition, the velocity distribution is given by a simple parabolic expression, according to Eqs. (3.1-3.2). As stated before, the eigenvalues and eigenfunctions of Eq. (5.23) have to be calculated numerically. This can be done with the method described in Appendix C. Laminar Pipe Flow

In order to investigate this case, the flow index F in all the equations has to be set to 1. This case has been investigated with the method of Papoutsakis et al. (1980a). Table 5.1 shows the first fifteen eigenvalues and constants for two different Peclet numbers. It can be seen that the positive eigenvalues strongly increase with growing values of the Peclet number. For example, the first positive eigenvalue ( λ0+ ) increases by about a factor of three by increasing the Peclet number from 5 to 10. Since larger eigenvalues result, according to Eq. (5.48), in a faster decrease of the influence of axial heat conduction, this elucidates how this effect diminishes with increasing values of the Peclet number. The increase of the positive eigenvalues with growing Peclet numbers can be seen clearly in Fig. 5.2, where the first 100 positive eigenvalues are plotted for four different Peclet numbers. Table 5.1: Eigenvalues and constants for laminar pipe flow for PeeD = 5 and PeeD = 10 (adapted from Papoutsakis et al. (1980a) and own calculations).

Pe D

5

Pe D

10

j

λ j+

A+j

λ j+

A+j

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

12.289076 18.929129 26.360733 34.041020 41.799450 49.593035 57.405409 65.229022 73.059822 80.895593 88.734839 96.576707 104.420491 112.265848 120.112354

-0.342609 0.412438 -0.374175 0.334422 -0.303680 0.279668 -0.260405 0.244561 -0.231254 0.219881 -0.210023 0.201373 -0.193702 0.186843 -0.180663

37.383579 51.950323 64.468355 78.791847 93.763713 109.009770 124.400395 139.876976 155.408680 170.978041 186.574029 202.189809 217.820419 233.462654 249.114013

-0.107926 0.232838 -0.303513 0.296627 -0.279034 0.262030 -0.247011 0.233958 -0.222598 0.212646 -0.203861 0.196042 -0.189034 0.182711 -0.176971

132

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

j

−λ j−

A−j

−λ j−

A−j

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

2.844835 10.174296 17.859425 25.617868 33.410361 41.221752 49.044572 56.874853 64.710161 72.549123 80.390694 88.234332 96.079491 103.925919 111.773292

1.233275 -0.617349 0.464093 -0.388494 0.340928 -0.307382 0.282080 -0.262122 0.245860 -0.232280 0.220720 -0.210726 0.201973 -0.194223 0.187302

3.372024 15.383971 29.751723 44.738343 59.986307 75.375199 90.849326 106.378881 121.946337 137.540929 153.155416 168.785154 184.426542 200.077324 215.735491

1.379949 -0.680027 0.493104 -0.406545 0.354040 -0.317634 0.290432 -0.269111 0.251826 -0.237452 0.225259 -0.214752 0.205575 -0.197472 0.190250

Fig. 5.2: Distribution of the positive eigenvalues for different Peclet numbers

Fig. 5.3 shows the temperature distribution along the pipe centerline for different Peclet numbers. It can be seen how the axial heat conduction effect in the fluid decreases rapidly with increasing values of the Peclet number. In addition, it can be seen from this figure that, for a Peclet number of 5, the region of influence of the axial heat conduction is about one radius in the negative direction. Fig. 5.4 shows the development of the temperature profile for different axial positions in the pipe (Papoutsakis et al. (1980a)). Note that, for growing values of the axial coordinate, the temperature tends to TW and thus Θ tends to zero. For x ≤ 0 the temperature of the fluid rapidly decreases to T0, resulting in values of Θ close to one for increasing distances from 0 . By comparing the two plots in Fig. 5.4, one can notice how the influence of the axial heat conduction in the flow

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

133

Fig. 5.3: Temperature distribution at the pipe centerline for different Peclet numbers (Papoutsakis, Ramkrishna and Lim (1980a))

Fig. 5.4: Temperature distribution for several axial positions x /( Pe D ) and two Peclet numbers (Papoutsakis, Ramkrishna and Lim (1980a))

134

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

gets weaker with increasing values of the Peclet number. For PeD = 20, the temperature distribution at the entrance of the heating section ( 0 ) becomes rather non-uniform. This shows clearly that whenever the axial heat conduction is not negligible, it is incorrect to prescribe a constant entrance temperature at 0 . In this case, the temperature distribution at 0 is always a result of the solution. Papoutsakis et al. (1980a) calculated the value of the Nusselt number for fully developed flow Nu ∞− , according to Eq. (5.54), and showed graphically the distribution as function of the Peclet number. In general, it can be seen that Nu ∞− increases with decreasing values of the Peclet number. Hennecke (1968) predicted numerically the value of Nu ∞− for several Peclet number. These values have been tabulated by Shah and London (1978). Table 5.2 shows a comparison between analytically and numerically predicted values of Nu ∞− . From Table 5.2, a very good agreement between numerically and analytically predicted values can be noted. Table 5.2: Comparison between numerically (Hennecke (1968), values taken from Shah and London (1978)) and analytically (values in the Table have been predicted by the author using the equations given by Papoutsakis et al. (1980a)) predicted values for Nu ∞− PeD 1 2 5 10 20 50

Nu ∞− (Papoutsakis et al. (1980a)) 4.027 3.922 3.767 3.695 3.668 3.659

Nu ∞− (Hennecke (1968)) 4.03 3.92 3.77 3.70 3.67 3.66

In Figs. 5.5–5.6, the distribution of the local Nusselt number is shown for laminar pipe flow for 0 . These values have been calculated using the numerical approach described in detail in Appendix C. In order to resolve the distribution of the Nusselt number for very small values of x correctly, a very large number of eigenvalues is needed. For the present calculations 200 eigenvalues have been taken. If less eigenvalues are used, the Nusselt number for 0 will tend to smaller values. This can be seen clearly, if the Nusselt number is plotted on a logarithmic scale. In addition to the large number of eigenvalues, a very fine grid was used in order to resolve the shape of the higher eigenfunctions (see Appendix C). Normally 1500 – 2000 grid points in the radial direction are sufficient. As it can be seen from the Figs. 5.5-5.6, the analytical method is in excellent agreement with the numerical calculations by Hennecke (1968) for the two shown Peclet numbers. For PeeD = 50, the analytical solution has also been compared with the investigations of Bayazitoglu and Özisik (1980) and of Singh (1958) and a very good agreement was found.

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

135

Fig. 5.5: Distribution of the local Nusselt number for x > 0 and PeeD = 1

Fig. 5.6: Distribution of the local Nusselt number for x > 0 and PeeD = 50

Laminar Channel Flow

In order to investigate this case, the flow index F in all the equations has to be set to 0. The first solution to this problem was obtained by Deavours (1974). He decomposed the eigenvalue problem, for the parallel plate channel, into a system of ordinary differential equations for which he proved the orthogonality of the eigenfunctions. As stated before, this case was also investigated by Weigand et al. (1993) with the analytical method presented here. Table 5.3 shows eigenvalues and constants for two different Peclet numbers. Similar to the circular pipe case (Table 5.1), it can be seen from Table 5.3 that the positive eigenvalues increase dramatically with growing values of the Peclet number. This shows again that the

136

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

effect of the axial heat conduction in the fluid diminishes with growing values of the Peclet number.The value of the Nusselt number for fully developed channel flow, Nu ∞− , has also been reported in literature. Table 5.4 shows a comparison between the analytical predicted values and calculations of Ash (see Shah and London (1978)) and Nguyen (1992). The values are ranging from PeeD = 0.02352 up to PeeD = 1000 and show a very good agreement between the presented analytical solution and literature data. It can be seen that the Nusselt number for the fully developed flow increases with decreasing values of the Peclet number. This behaviour is very similar to the results shown in the previous section for the circular pipe. Table 5.3: Eigenvalues and constants for laminar channel flow (PeeD = 5 and PeeD = 10)

Pe D + j

5

Pe D + j

10 A+j

j

λ

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

3.220469 6.758122 10.645334 14.557251 18.476438 22.398690 26.322503 30.247224 34.172509 38.098175 42.024105 45.950232 49.876499 53.802882 57.729349

-0.376403 0.202738 -0.124312 0.089495 -0.069885 0.057315 -0.048575 0.042147 -0.037220 0.033325 -0.030167 0.027556 -0.025361 0.023489 -0.021875

9.575077 15.522013 23.101943 30.850355 38.648148 46.467380 54.297781 62.134735 69.975845 77.819763 85.665658 93.513006 101.361442 109.210726 117.060670

-0.230822 0.192715 -0.121091 0.087928 -0.068971 0.056719 -0.048157 0.041837 -0.036981 0.033135 -0.030013 0.027428 -0.025253 0.023398 -0.021796

j

−λ j−

A−j

−λ j−

A−j

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1.192001 5.139142 9.062116 12.984172 16.907540 20.831909 24.756940 28.682423 32.608218 36.534242 40.460433 44.386753 48.313173 52.239673 56.166236

0.897756 -0.222463 0.130123 -0.092272 0.071518 -0.058393 0.049341 -0.042719 0.037664 -0.033679 0.030456 -0.027797 0.025564 -0.023664 0.022026

1.580670 8.989775 16.748478 24.547406 32.366159 40.195993 48.032482 55.873251 63.716909 71.562614 79.409807 87.258131 95.107316 102.957190 110.807606

1.044165 -0.232941 0.132737 -0.093479 0.072234 -0.058873 0.049686 -0.042979 0.037868 -0.033843 0.030592 -0.027910 0.025661 -0.023747 0.022098

A

λ

+ j

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

137

Table 5.4: Nusselt number for the fully developed flow , Nu ∞− , compared to predictions by Ash (see Shah and London (1978)) and Nguyen (1992). PeD

0.02352 0.4444 0.6508 1.0576 1.4368 2 5 9.97 50 69.78 100 1000

Nu ∞− (Eq. (5.54)) 8.1141 8.0644 8.0416 7.9997 7.9640 7.9165 7.7471 7.6310 7.5457 7.5432 7.5419 7.5407

Nu ∞−

Nu ∞−

Ash (see Shah and London (1978)) 8.1144 8.0644 8.0416 7.9998 7.9640

Nguyen (1992)

7.9164 7.7468 7.6310 7.5456 7.5408 7.5407 7.5407

The temperature distribution at the centreline of the fluid is depicted in Fig. 5.7. By comparing Fig. 5.7 with Fig. 5.3 for the circular pipe, it can be seen that the axial heat conduction effect within the fluid has very similar effects for the two geometries.In addition, it can be seen that for very low values of the Peclet number the temperature at 0 is lowered because of axial heat conduction effects within the flow.

Fig. 5.7: Temperature distribution at the centerline of the parallel plate channel for different Peclet numbers

Fig. 5.8 shows the distribution of the local Nusselt number for PeeD = 4 and 0. The Nusselt number is compared to calculated values of Agrawal (1960). Agrawal 0 two independent series solutions, which (1960) determined for 0 and

138

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

then have been matched at 0 . This approach resulted in a complicated expression for the Nusselt number. From Fig. 5.8, it can be seen that both calculations are in good agreement for 0.1 . However, for smaller values of x , the values given by Agrawal (1960) are much lower than the Nusselt numbers predicted here. The reason for this behavior is caused by the fact that Agrawal used only five eigenvalues for the prediction of the local Nusselt number. This is sufficient for larger values of the axial coordinate, but not for very small values of the axial coordinate2. For very small values of the axial coordinate, it can be seen from Fig. 5.8 −β that the Nusselt number can be approximated by Nu D ∝ x . This is in good agreement with predictions by Hennecke (1968) and by Kader (1971).

Fig. 5.8: Distribution of the local Nusselt number for Pe D

4(

0)

5.1.2 Heat Transfer in Turbulent Pipe and Channel Flows for Small Peclet Numbers

For the heat transfer in a turbulent duct flow, the functions a1 and a2 depend on the turbulent heat exchange in the flow. For the following analysis, the ratios hx / ε hr and ε hx / ε hy are set equal to one. This assumption has been made previously by Lee (1982) and Chieng and Launder (1980). Weigand et al. (2002) proved this assumption by solving Eq. (3.6) numerically by using a non-isotropic heat flux model for the heat transfer in a planar channel. They concluded that the above assumption is well justified for the applications considered here. Assuming that the 2

If only the first five eigenvalues for the prediction of the Nusselt number are used with the present method, the predicted values are in close agreement with the ones by Agrawal (1960) (see Fig. 5.8).

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

139

ratio ε hx / ε hn is equal to one, results in the fact that a1 is equal to a2 (see Eq. (5.8)). Since axial heat conduction effects in turbulent flows are only important if the Prandtl number is very small, the turbulent Prandtl number has to be calculated by adopting a model suitable for liquid metal flows (e.g. Eqs. (3.79-3.82)). Turbulent Pipe Flow

If the flow index F is set to one in the equations, the turbulent heat transfer in a pipe for a low Peclet number flow can be predicted. For the following calculations the turbulent Prandtl number model of Azer and Chao (1960) (see Eq. (3.82)) as well as the Kays and Crawford model (1993) (see Eqs. (3.79-3.81)) have been used. These two models have been found to give reliable answers for turbulent liquid metal flows. The reader is referred to Weigand (1996) or to Weigand et al. (1997) for comparisons between calculations and measurements for liquid metal flows in pipes with the above mentioned models for Prt. Table 5.5: Eigenvalues and constants for turbulent pipe flow with Pr = 0.001.

Re D j

λ

0 1 2 3 4 5 6 7 8 9

+ j

5000

Re D + j

+ j

10000 A+j

A

λ

10.915 17.925 25.530 33.266 41.051 48.859 56.628 64.512 72.348 80.189

-0.3959 0.4220 -0.3748 0.3354 -0.3049 0.2810 -0.2617 0.2458 -0.2324 0.2209

33.569 45.293 59.584 74.598 89.896 105.33 120.85 136.41 152.01 167.63

-0.2064 0.3225 -0.3210 0.3018 -0.2820 0.2644 -0.2491 0.2359 -0.2244 0.2144

j

−λ j−

A−j

−λ j−

A−j

0 1 2 3 4 5 6 7 8 9

3.305 10.626 18.337 26.127 33.947 41.779 49.617 57.460 65.305 73.152

1.2055 -0.6417 0.4758 -0.3936 0.3430 -0.3080 0.2820 -0.2618 0.2453 -0.2317

4.280 16.837 31.444 46.622 62.023 77.532 93.102 108.71 124.34 139.98

1.3923 -0.7376 0.5281 -0.4262 0.3653 -0.3242 0.2943 -0.2713 0.2530 0.2380

Eigenvalues and constants are shown in Table 5.5 for a Prandtl number of 0.001 and different values of the Reynolds number. It can be seen from Table 5.5 that the positive eigenvalues increase rapidly with growing values of the Reynolds

140

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

number. This shows again the decreasing influence of the axial heat conduction within the flow. Fig. 5.9 shows the distribution of the Nusselt number for fully developed flow ( Nu∞− ) as a function of the Peclet number for Pr = 0.022. The predicted values have been obtained by using different models for the turbulent Prandtl number. It can be seen that the two models give relatively similar results. The predicted values have been compared to experimental data from Sleicher et al. (1973) and from Gilliland et al. (1951) (taken from Azer and Chao (1960)). As it can be seen from Fig. 5.9, the agreement between experimental data and predictions is good. Table 5.6 shows values of the Nusselt number Nu ∞− for fully developed flow for different values of the Prandtl and Reynolds number. In addition, the table contains the relative deviation of the Nusselt number for a parabolic calculation ΔE = ( ) / Nu elliptic .

Fig. 5.9: Nusselt number of the fully developed flow ( Nu ∞− ) as a function of the Peclet number for Pr = 0.022 (Weigand, Ferguson and Crawford (1997))

From the entries in Table 5.6, it is obvious that axial heat conduction effects are not important for the fully developed flow. Therefore, the influence of axial heat conduction on the Nusselt number for fully developed flow can be neglected with good accuracy. This may not be the case when the thermal entrance region is considered. Fig. 5.10 shows the relative error ΔE in the local Nusselt number by ignoring the axial heat conduction within the thermal entrance region. As mentioned before, the relative error is negligible for the fully developed flow ( x → ∞ ). However, it may be very large for small values of the axial coordinate. For example for PeeD = 10, the error will be about 30% for /( Pe D ) 0.01 and for smaller values of the axial coordinate this error is even larger.

5.1 Heat Transfer for Constant Wall Temperatures for x d 0 and x > 0

141

Table 5.6: Nusselt number for fully y developed p flow ( Nu ∞− ). The values in brackets indicate the relative deviation ΔE = / Nu elliptic between an elliptic and a parabolic calculation.

(

Pr

)

0.002

0.006

0.01

8.370 (0.014) 8.518 (0.007) 8.760 (0.002) 8.919 (0.0004) 9.012 (0.0002)

8.284 (0.003) 8.484 (0.001) 8.783 (0.0002) 9.011 (0.0) 9.175 (0.0)

8.277 (0.001) 8.497 (0.0004) 8.835 (0.0001) 9.149 (0.0) 9.408 (0.0)

ReD 3000 5000 10000 20000 30000

Fig. 5.10: Relative error in the local Nusselt number by neglecting the axial heat conduction within the fluid (Weigand (1996))

This example shows very clearly that axial heat conduction effects may be important in the thermal entrance region, also for turbulent flows. Fig. 5.11 shows the local temperature field in the flow for two different values of the Peclet number. The increasing effect of the axial heat conduction in the flow for decreasing values of the Peclet number is clearly visible for small axial values.

142

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

Fig. 5.11: Temperature distribution for a thermally developing flow in a circular pipe (Weigand (1996))

Turbulent Flow in a Planar Channel

For flow and heat transfer in a parallel plate channel, the flow index F in the preceeding equations must be set to 0. The behaviour of the solutions, e.g. for the Nusselt number, is very similar to the one for the circular pipe and therefore is not been presented here. The reader is referred to Weigand (1996) and Weigand et al. (2002) for a detailed presentation of these results.

5.2 Heat x>0

a s e for Constant Wall Heat Flux for x ≤ 0 and

This case has been studied analytically by Papoutsakis et al. (1980b) for laminar, hydrodynamically fully developed pipe flow and by Weigand et al. (2001) for laminar and turbulent hydrodynamically fully developed pipe and channel flows.

5.2 Heat Transfer for Constant Wall Heat Flux for x d 0 and x > 0

143

The geometry under consideration and the used coordinate system are shown in Fig. 5.12

Fig. 5.12: Geometrical configuration and coordinate system

Similar to the considerations given in Chap. 5.1, we start with Eq. (5.3)

ρ cp u (

)

∂T ∂x

1 ∂ ª F «r k F ∂n ¬ ©

cp

εm T º Prt ¹ n ¼

ª x ¬©

k

cp

· ∂T º » Prt ¹ ∂xx ¼ P m

(5.3)

This equation has to be solved with the boundary conditions n = 0 : ∂ / ∂ = 0, n = L: ≤ 0: ∂ /∂ x 0 : ∂T / ∂n

(5.55) = 0, qW / k

lim T T0

x → −∞

Introducing the following dimensionless quantities Θ=

(5.56)

T T0 x 1 u n r , x = , u= , n= , r = qW L / k L Pe L u L L

Pe L Re L Pr Pr, Re L

uL

ν

, Pr

ν a

, m

ε ε , Pr Pt= m ν ε hn

into the energy equation (5.3) and into the boundary conditions, Eq. (5.55), results in u (

)

∂Θ 1 ∂ ª = 2 a1 ( ∂x P Pee L ∂x «¬

with the boundary conditions

)

∂Θ º 1 ∂ ª F + r a2 ( ∂x »¼ r F ∂n «¬

)

∂Θ º ∂n »¼

(5.6)

144

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

x → −∞ : Θ = 0 ∂Θ n = 0 : 0 ∂n ∂Θ n 11:  0 : ∂n

(5.57)



1,,

0:

∂Θ ∂n

0

Similar to the procedure shown in Chap. 3.3, the temperature for the present case has been made dimensionless with the constant wall heat flux and the thermal conductivity of the fluid. The functions a1 and a2 are given by Eq. (5.8). The solution procedure for the energy equation for a constant heat flux at the wall is similar to the one for constant wall temperature. We are introducing again the axial energy flow through a cross-sectional area of the height n ( ( , ) ), given by Eq. (5.9). This results again in G a system off partial differential equations, Eq. (5.10), with the solution vector S and the matrix operator L defined by Eq. (5.11). The  boundary conditions for the function ( , ) can be derived from the definition of the axial energy flux, Eq. (5.9) by using Eq. (5.57). One obtains lim

0 ,

x → −∞

n

,

­0, 1: E = ® , ¯ x,

0:

(5.58)

0

−∞ < x ≤ 0  0

For the inner product, the expression given by Eq. (5.14) can be used again. However, because of the changed boundary conditions, D ( ) needs to be adapted G G (5.59) D(( ) : (exists and) , 2 (1) 2 (0) 0   where the only change in Eq. (5.59) compared to Eq. (5.15) lies in the changed boundary condition that Φ 2 (1) 0 . For the eigenvalue problem one obtains again

{

}

ª¬ r F a2 (n)

j11

º¼′ + r F

ª λ j a1 ( ) º − u λ j Φ j1 = 0 2 ¬ PeL ¼

(5.23)

but now with the boundary conditions Φ ′j1j1 (0) 0

, ′j1 (1) 0

(5.60)

Comparing the boundary conditions given by Eq. (5.60) for a constant wall heat flux to the ones for a constant wall temperature, it can be seen that only the boundary condition for 1 has been changed. The normalizing condition for the eigenfunctions, Eq. (5.25), can be used again. Similarly as for Eq. (5.26) it can be G shown that an arbitrary vector f can be expanded into a series of eigenfunctions and the result is given by Eq. (5.30).

5.2 Heat Transfer for Constant Wall Heat Flux for x d 0 and x > 0

145

Since the expansion of an arbitrary vector in terms of eigenfunction has already been developed, we focus directly on the solution of the system of partial differential equations given by Eq. (5.10). Because of the non-homogeneous G boundary condition for the heat flux, the vector S does not belong to the domain D ( ) given by Eq. (5.59). However, it can be shown that G G L S,, 

G j

G

,

j

j1

(

(1)

)

(5.61)

Note here the difference in this equation compared to Eq. (5.31). The last term on the right hand side of Eq. (5.61) is now a function of the axial coordinate and no longer a constant. Taking the inner product of both sides of Eq. (5.61) one obtains ∂ G G S,, ∂xx

λ

j

G G ,

j

j

j1

(1))

(

(5.62)

)

G G Eq. (5.62) represents an ordinary differential equation for the expression S , Φ j and can be solved separately for positive and negative eigenvalues. This results in G G S,

x

(

,1) exp((λ j (

(

,1) exp((λ j+ (

j1 (1) ³

j

(5.63)

)) dx

−∞

G G S,, +j

+ j1

x

(1) ³

−∞

(5.64)

)) dx

After evaluating the integrals in Eqs. (5.63-5.64), the following result for the temG perature distribution Θ, which is the first vector component of S , can be derived Φ +j1 ( ) exp ( G+ 2 2 j =1 Φ λ +j j ∞

)=¦

x ≤ 0 : Θ (

Φ j1j1 (

∞

x > 0 : Θ (   ) = ¦ j 0

∞

j =0

Φ

j1 j1

2

λ j2

−¦

)

( n)

()

λ

−2 j

j

G Φ −j

( ) 2

+ j1

j1

x j 0

j − j1 j1

)Φ

(5.65)

( )

()

λj Φ j

2

Φ j1 (

)

(5.66)

exp(λ j )

The temperature distribution given above contains for 0 both negative and positive eigenfunctions. The expressions for the temperature field can be further simplified by replacing the first two terms in Eq. (5.66). This results (see Appendix D) in

146

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

0 : Θ ( x , n )

x

with the function Ψ(

n

)=³ 0

∞

¦

0 : Θ ( ,  )

x

+ j

j =0

eexpp (

Ψ((n ) ( F 1)) x

+ j

)

+ j1

∞

¦A

j

j1

j =0

( )

(5.67)

( n ) exp((λ j

)

(5.68)

( ) given by F +1 r a2 ( F

n

u ( ) r )³

F

dds ddn

Ψ (n)

C2

C2

(5.69)

0

where C2

(

)

2 1

Pe 2L

F ³ r a1 (

)

(

0

1

)³

( ) Ψ (  )ddn

F

(5.70)

0

The coefficients Aj are defined by Aj =

Φ j1j1 (

)

2

λj Φ j

2

ª = −1/ «λ j ¨ « dλ © ¬

j1

º » ¸¸ ¹λ λ j ¼»

( )·

λ

(5.71)

It is interesting to focus again on the differences in the solutions for the temperature field for the case of a constant wall temperature and for the case of a constant wall heat flux. For the case of a constant wall temperature, Eqs. (5.40-5.41) show that the temperature in the field changes like an exp-function for positive and negative values of the axial coordinate. Physically, this means that the fluid tries to achieve asymptotically the changed wall temperature after the temperature jump at 0 . For the case of a constant heat flux at the wall, the temperature of the fluid will rise with increasing values of the axial coordinate. This is reflected by the second term on the right hand side of Eq. (5.68). The third term in Eq. (5.68) represents the response of the system to the change in the heat flux for x = 0 , whereas the first term on the right hand side in Eq. (5.68) represents the fully developed temperature distribution, for large values of the axial coordinate. This behaviour is very similar to the one already discussed in Chap. 3.3. However, there is an important difference, which lies in the fact that the inlet temperature profile (at 0 ), for the case with axial heat conduction, is now part of the solution and no longer a boundary condition. After the temperature distribution in the flow is known, the distribution of the bulk-temperature and the Nusselt number can be calculated from their definitions given by Eqs. (3.64-3.65). This results in x

0 : Θb ( x )

(F

∞

) ¦ A+j eexpp ( j =0

1

+

x ) ³ ur F 0

+ j1

d ( n )dn

(5.72)

5.2 Heat Transfer for Constant Wall Heat Flux for x d 0 and x > 0

x

(

0 : Θb ( x )

³r

Pe 2L

−¦ Aj exp e p(

(

a1 ( n )ddn

1

j

j =0

4

F

(F

) ³ ur

F

ª ¬

(5.73)

j1

( n ) ddn +

0

2 ­° ( ) 1 F r a( ® ) °¯ Pe2L ³0 1

) «(

)x

0

∞

x > 0 : N Nu D

j =0

2 1

e p( ) ¦ Aj exp

−(

∞

)

147

)

1 F j

)

( )

0

()

(5.74)

º½

j

( ) » °¾ ¼ ¿°

Because of the applied boundary conditions of an adiabatic wall for Nusselt number has to be zero in this region.

0 , the

5.2.1 Heat Transfer in Laminar Pipe and Channel Flows for Small Peclet Numbers

For a laminar flow, the functions a1 and a2 are equal to one. In addition, the velocity profile is given by a simple parabolic distribution, see Eqs. (3.1-3.2). For this simple case, the function ( ) , appearing in the temperature distribution, can be predicted analytically (see e.g. Papoutsakis et al. (1980b) for pipe flow). For large values of the axial coordinate ( x → ∞ ), the temperature distribution for 0, Eq. (5.68), therefore simplifies to Pipe

r 4 2 7 + − 4 Pe 2L 24

(5.75)

3 2 y 4 1 1 y − + 2 − 4 12 eL 7

(5.76)

Θ( , ) = 2 x + r 2 −

Planar Channel

Θ( , y ) = x +

which is the temperature distribution of the fully developed flow.

148

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

Laminar Pipe Flow

In order to investigate this case, the flow index F in all the equations has to be set to one. This has been investigated by Papoutsakis et al. (1980b) for the case of a semi-infinite and also for a finite heated zone. Although they derived in their paper equations for the local Nusselt number distribution, only temperature distributions have been shown in their paper. Fig. 5.13 shows the distribution of the bulktemperature for various values of the Peclet number. This figure elucidates the effect of axial heat conduction. It can be seen that for PeeD = 5, the bulk-temperature is already zero for small distances upstream of 0 . This is different for smaller Peclet numbers. For PeeD = 1, one notices the large effect of the axial heat conduction in the fluid, leading to finite values of Θb also far upstream of the entrance into the heating section. The analytically predicted values of the bulk-temperature have been compared with numerical predictions of Hennecke (1968) and very good agreement is found.

Fig. 5.13: Distribution of the bulk-temperature for different values of the Peclet number (Weigand, Kanzamar and Beer (2001))

Fig. 5.14 shows comparisons between present calculations and predictions by Hsu (1971a), Nguyen (1992), Bilir ((1992) and Hennecke (1968) for two different values of the Peclet number. As it can be seen from Fig. 5.14, the agreement between the present analytical calculations and the Nusselt number from other predictions is very good. Fig. 5.15 shows the distribution of the local Nusselt number for five different Peclet numbers (data taken from Nguyen (1992)). As it can be seen from Fig. 5.15, the curves for the Nusselt numbers have an inflection point, i.e. NuuD increases with decreasing Peclet numbers. This phenomenon has been found by various workers, e.g. Hennecke (1968) and Hsu (1971a).

5.2 Heat Transfer for Constant Wall Heat Flux for x d 0 and x > 0

149

Fig. 5.14: Distribution of the local Nusselt number for two different Peclet numbers

Fig. 5.15: Local Nusselt numbers for pipe flow ( qW = const. ) and various PeeD (data taken from Nguyen (1992))

150

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

Laminar Channel Flow

In this case, the flow index F has to be set to zero in all equations. Fig. 5.16 shows typical temperature distributions ΘW − Θ for various axial locations. One clearly sees that already for 0 a temperature profile exists because of the axial heat conduction effects within the fluid. This profile is also present upstream of the start of the heating section. For larger values of the axial coordinate the fully developed temperature distribution, given by Eq. (5.75), is obtained asymptotically.

Fig. 5.16: Temperature distribution ΘW − Θ for PeeD = 5 and various axial positions

5.2.2 Heat Transfer in Turbulent Pipe and Channel Flows for Small Peclet Numbers

For a turbulent flow, the functions a1 and a2 are depending on the turbulent heat exchange in the flow. For the calculations, assuming a constant wall heat flux boundary condition, the ratio hx / ε hn can be set to one and the models for the turbulent Prandtl number, explained in paragraph 5.1.2, can be used. Turbulent Pipe Flow

The general behavior for turbulent pipe flow is similar to the one for laminar flow, even though radial mixing process is enhanced by the turbulent fluctuations. Fig. 5.17 shows distributions of the centerline temperature in the pipe for three different Peclet numbers and a Prandtl number of 0.001. For a Peclet number as high as 100, it can be seen that no heat is transferred upstream of 0 . Therefore, this problem could be treated as parabolic. For lower values of the Peclet number, axial heat conduction effects start to change the centerline temperature upstream of x = 0 . Because of the heat transferred upstream, the fluid temperature at the cen-

5.2 Heat Transfer for Constant Wall Heat Flux for x d 0 and x > 0

151

terline increases to values larger than zero. Fig. 5.18 shows a comparison between analytically predicted Nusselt numbers for the fully developed flow ( Nu ∞− ) and experimental data of Fuchs (1973). It can be seen that the predictions are in good agreement with the experimental data.

Fig. 5.17: Temperature distribution at the pipe centerline for Pr = 0.001

Fig. 5.18: Nusselt number for fully developed flow (Nu ) as a function of the Reynolds number (adapted from Weigand, Kanzamar and Beer (1997))

152

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

Fig. 5.19 shows the distribution of the local Nusselt number for different Peclet numbers and Pr = 0.001. It can be seen, that increasing values of the Peclet number result in enhanced turbulent mixing and, therefore, in increased values of the Nusselt numbers.

Fig. 5.19: Variation of the local Nusselt number for different Peclet numbers and Pr = 0.001 (Weigand et al. (2001))

Turbulent Channel Flow

For this case, the flow index F has to be set to zero in all equations. Table 5.7 gives the first ten eigenvalues and constants for the turbulent flow in a parallel plate channel for Pr = 0.001 and for two different values of the Reynolds number. Table 5.7: Eigenvalues and constants for turbulent channel flow with Pr = 0.001.

Re D

5000

Re D

10000

j

λ j+

A+j

λ j+

A+j

0 1 2 3 4 5 6 7

0 100.72 179.41 197.87 223.14 255.70 290.88 327.40

0 2.9632E-4 -2.0154E-3 4.1893E-3 -3.9981E-3 2.9457E-3 2.2146E-3 1.7109E-3

0 353.45 622.30 690.36 729.86 766.13 817.95 878.26

0 4.0451E-7 -1.9948E-05 2.7807E-04 -9.6244E-04 1.3601E-03 -1.2109E-03 1.0252E-03

8 9 10

36469 402.47 440.58

-1.3554E-3 1.0973E-3 -9.0511E-4

943.31 1011.5 1081.8

-8.7132E-04 7.4558E-04 -6.4275E-04

5.3 Results for Heating Sections with a Finite Length

153

j

−λ j−

A−j

−λ j−

A−j

0 1 2 3 4 5 6 7 8 9 10

0 10.31 34.50 65.28 99.21 134.85 171.53 208.88 246.68 284.80 323.14

0 -1.7544E-1 4.2981E-2 -1.8317E-2 9.8721E-3 -6.0627E-3 4.0512E-3 -2.8746E-3 2.1344E-3 -1.6420E-3 1.2997E-3

0 10.82 40.14 83.48 136.53 196.11 260.15 327.31 396.72 467.82 540.18

0 -1.8372 E-1 4.7222 E-2 -2.0737E-2 1.1391E-2 -7.0954E-3 4.7916E-3 -3.4235E-3 2.5503E-3 -1.9623E-3 1.5496E-3

As it can be seen from the entries in Table 5.7, the first nonzero negative eigenvalue (λ1 ) stays nearly constant for both Reynolds numbers. On the other side, it can be seen that the first nonzero positive eigenvalue (λ1+ ) increases very strongly with growing values of the Reynolds number. This shows again the decreasing importance of the effect of axial heat conduction with growing values of the Peclet number. This behavior is very similar to the laminar flow case considered before.

5.3 Results for Heating Sections with a Finite Length In the previous sections, we investigated the axial heat conduction effect for a hydrodynamically fully developed flow for a semi-infinite heating section. For this type of applications, the influence of the axial heat conduction scales very well with the Peclet number, and for flows with Peclet number larger than 100 axial heat conduction effects can be neglected with good accuracy. Furthermore, for the hydrodynamically and thermally fully developed flow, the effect of axial heat conduction can normally be neglected with sufficient accuracy. However, the above given guidelines may not be applicable, if the length of the heating section decreases in size. It is quite obvious that axial heat conduction influences the heat transfer in a very short heated section also for larger values of the Peclet number. In addition, it should be noted that this type of problems can not be calculated correctly using the simplified energy equation (e.g. Eq.(3.14)), which is parabolic in nature. This is obvious because the parabolic nature of the simplified energy equation permits that the end of the heating zone influences the conditions within the heating section. The present section explains how the analytical method previously established can be easily adapted to this sort of problems. The resulting solution is as simple and efficient to compute as the previously discussed solutions for the semi-infinite heating section.

154

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

5.3.1 Piecewise Constant Wall Temperature

This case has been recently solved by Lauffer (2003) and Weigand and Lauffer (2004). The geometry under investigation is shown in Fig. 5.20. It can be seen that the wall temperature changes from T0 to TW for 0 < x < x1 . The length of the heating section is x1.

Fig. 5.20: Geometry and boundary condition for a piecewise constant wall temperature

The analysis presented in Chap. 5.1 can be used to solve the present problem. However, the boundary conditions have to be adapted and now are given by n L : T T0 , x 0, x T TW , 0

x1

(5.76)

x x1

n =0 : ∂ / ∂ =0 lim lim T T0 o , x → −∞

x →+∞

The changed boundary conditions result in a changed expression for the function g ( ) , which appears in Eqs. (5.36-5.37). If the integrals in Eqs. (5.36-5.37) are solved, the following temperature distribution in the fluid can be obtained (Lauffer (2003)) ∞

x 0:

( , ) 1

¦ j =0

+ j

+ j1

( ) exp((λ j ) ª¬1 exp(( λ +j 1 ) º¼

(5.77)

5.3 Results for Heating Sections with a Finite Length

0



1 : ∞

¦

Θ( , )

j

j1

j 0

) ¦A

(

¦

) 1

j

j1

j =0

x1 / (

where x1

).

+ j

+ j1

j 0

∞

x x1 :

(5.78)

∞

( ) exp (

155

( ) exp (

( n) exp e p(

) ª¬1

))

(

exp(( λ j

1

) º¼

(5.79)

From the above equations, it can be seen that, for

x1 → ∞ , the results for the semi-infinite heating section are obtained, (see Eqs. (5.48-5.49)). Furthermore, it is interesting to note that for the case of a finite heated section the temperature distribution within the heating zone is influenced by positive and negative eigenvalues and eigenfunctions (see Eq. (5.78)). The distribution of the Nusselt number can be obtained from the definition of this quantity and the known temperature distribution given by Eqs. (5.77-5.79). This results in ∞

−4¦ j=0

x ≤ 0 : Nu D = 4

F

∞

1

¦ ³ j

j =0

0



j1

) )º¼

(

1 :

Nu D =

−4¦

(5.81)

∞

4¦

′ (1)

+ j

+ j1

j 0

′j1 (1) ( ) eexp( p((λ j )

j

j 0

N ∞

1

4F ¦

j

j =0

∞

³0

j1 (

j

j =0

³0

j1 (

∞

−4¦ ∞

j =0

F

)

1

j

³ 0

)) +

(

exp λ +j ( exp e p

(

j1

( )

F

(5.82)

)

′j1 (1) ª exp((λ j ) exp e p( ¬

j

j =0

4F ¦

F

)

1

4F ¦

x ≥ x1 : Nu D =

(5.80)

) ) º¼

(

+ ªexp( e p( ¬e p((λ j ) exp

F

( )

0

∞

N

′j1+ (1) ª exp((λ +j ) exp e p( ¬

+ j

) )º¼

(

ªexp( e p( ¬e p((λ j ) exp

(

(5.83)

) ) º¼

The above equations for the distribution of the Nusselt number are identical to the ones given by Eqs. (5.52-5.53) for the case of a semi-infinite heating zone ( x1 → ∞ ).

156

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

Fig. 5.21: Comparison between numerically and analytically predicted distribution of the bulk-temperature for a turbulent flow in a parallel plate channel with turbulent internal flow (ReeD = 40000, Pr = 0.01, x1 = 0.1 )

As mentioned before, the problem of a piecewise heated planar channel has been solved numerically by Weigand et al. (2002). Fig. 5.21 shows a comparison between the analytically and numerically predicted distribution of the bulktemperature (Lauffer (2003)) for a turbulent flow and a length of the heated zone of x1 = 0.1 . It can be seen that the bulk-temperature tries to approach the wall temperature within the heated section. At the end of the heated section, the bulktemperature decreases then towards the uniform temperature T0. If the axial coordinate is scaled by the length of the heating zone ( xˆ x / x1 ), the effect of a changing length of the heating zone can be seen nicely. Fig. 5.22 shows the relative error between a parabolic and an elliptic calculation for PeeD = 10 and a laminar flow through a circular pipe. It can be seen that near the start and the end of the heating section always a large deviation between the elliptic and the parabolic calculation exists. In these areas, the axial heat conduction in the fluid changes the temperature distribution substantially in the pipe and therefore the values of the Nusselt number are changed. If the heating zone decreases in length, axial heat conduction might be important throughout the full heated section. For such a situation, the complete energy equation has to be taken into account. This is clearly visible in Fig. 5.22. For the short heating section of x1 = 0.1 , the axial heat conduction in the flow results in a dramatic change of the temperature field, resulting in about 40% relative error in the Nusselt number, even in the middle of the heated section.

5.3 Results for Heating Sections with a Finite Length

157

Fig. 5.22: Relative error of the Nusselt number between elliptic and parabolic calculations for a laminar pipe flow (PeeD = 10) with different length of the heating section

5.3.1 Piecewise Constant Wall Heat Flux

The case of a piecewise constant wall heat flux has been discussed by Papoutsakis et al. (1980b) and Weigand et al. (2001). The solution approach is very similar to the one outlined above for the piecewise constant wall temperature. The geometry and the boundary conditions under investigation are shown in Fig. 5.23.

Fig. 5.23: Geometrical configuration and boundary conditions

The boundary conditions given by Eq. (5.57) have to be changed according to

158

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

n = 0 : ∂ / ∂ = 0, n = L : ≤ 0, ≥

1

: ∂ /∂

(5.84)

= 0,

0 < x < x1 : ∂T / ∂n = qW / k lim T T0

x → −∞

This results in a changed description of the boundary conditions for E. This means that Eq. (5.58) has to be replaced by lim

0 ,

x → −∞

n

,

This changed expression for E (

(5.85)

0

−∞ < x ≤ 0 0 x x1

­0, ° ® , ° x ¯ 1

1:: 1

0:

x1 ≤ x < ∞

)

has to be inserted into the integrals in Eqs.

(5.63-5.64). These are, however, all the changes we need to do, in order to solve this relatively complicated problem. Evaluating Eqs. (5.63-5.64) results in the following temperature distribution for this case (Weigand et al. (2001)) Φ +j1 ( ) exp ( G+ 2 2 j =0 Φ λ j+ j

0

1 : Θ (  ,  )



( ) (

∞

−¦ A+j Φ +j1 ( j =0

x

x1 : Θ (

) ª1

∞

)=¦

x ≤ 0 : Θ (

) (

with the function Ψ (

) )

¬

j

j

j

j =0

( (

∞

)º¼ Φ +j1 (

∞

¦

1) 

)

¦

e p( exp

(  ) exp (

)

(5.87)

))

e p( ( ) ª¬exp

j

j =0

(5.86)

)

)

(

º ¼

)

(5.88)

given by Eq. (5.69). Again the distribution of the bulk-

temperature and the Nusselt number can be calculated from the definitions of these quantities and one finally obtains x

0 : Θb ( x )

∞

) ¦ A+j ª¬1

(F

0<
0. Both, the temperature distribution and the temperature gradient were then matched at x = 0 by constructing a pair of orthonormal functions from the eigenfunctions and using the Gram-Schmidt orthonormalization procedure. This method, therefore, was quite cumbersome for finding the solution off the problem. Compared to this solution, the present approach presents the analytical solution in a form, which is as simple as the one for the parabolic problem.

Fig. 5.25: Local Nusselt number for a concentric annuli with constant heat flux at the wall (Weigand and Wrona (2003))

For more details and also for applications for turbulent flows inside concentric annuli the reader is referred to Weigand et al. (1997) and Weigand and Wrona (2003). Recently, the above described method has also been used for the prediction of the thermally developing laminar flow of a dipolar fluid in a duct by Akyildiz and Bellout (2004). Dipolar fluids are the simplest example of a class of nonNewtonian fluids called multipolar fluids (see Bleustein and Green (1967) for more detailed information on this subject)). Problems

5-1 Consider the flow and heat transfer in a planar channel with height 2h. The plates have a constant wall temperature T0 for 0 and TW for 0 . The velocity profile of the laminar flow in the planar duct can be simplified to a slug flow profile with u u = const. . The velocity component in the y – direction v = 0. All fluid properties can be considered to be constant. Axial heat conduction in the flow can not be neglected, because of small values of the Peclet number. The temperature distribution in the fluid can be calculated from the energy equation

162

5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)

u

2 § 2T ∂T T· = a¨ 2 + 2 ¸ ∂xx ∂y ¹ © ∂x

with the boundary conditions x

0:T

y

∂T 0: ∂yy

T0 00,

y

h : T = TW

a.) Introduce the following dimensionless quantities x

x a h uh

T TW x 1 y , y= , Θ= hP Pee h T0 TW

What is the resulting differential equation and the boundary condition in dimensionless form? b.) Show that the above considered problem reduces in case of negligible axial heat conduction in the fluid ( Pe → ∞ ) to Problem 3.3. c.) Use the method of separation of variables to solve the given problem ( f ( ) g ( y ) ). What equations are obtained for the functions f and g? d.) Predict the complete solution for the temperature field. e.) Compare the obtained temperature distribution with the one of Problem 3.3 for three different Peclet numbers (Pe = 0.1, 1, 2) for x/h = 1. What influence has the axial heat conduction on the temperature profile? f.) Was it physically correct to prescribe a constant temperature distribution for x = 0? Explain your answer. 5-2. Problem 5-1 should now be solved with the method outlined in Chap. 5. The boundary condition for x 0 ::T T T0 is replaced by lim

x →−∞

0

, lim T x →∞

TW

a.) Introduce the dimensionless quantities given in Problem 5.1. What is the resulting dimensionless differential equation and what are the boundary conditions? b.) Split the problem into two first order partial differential equations by inG troducing a solution vector S consisting of the temperature and the axial energy flow. What system of differential equations is obtained? Show the associated eigenvalue problem and obtain its solution. c.) Determine the complete solution of the problem. d.) Compare your solution to the one of Problem 5.1 for x/h = 1 for different Peclet numbers. Explain your observations.

5.4 Application of the Solution Method to Related Problems

163

5-3. Consideration is given to the heat transfer in a concentric annuli which has constant temperatures at the inner and outer radii for 0 and a temperature jump for 0 at the outer radius. Show from the definition of the inner product (Eq. (5.93)) that G G Φ, Lϒ = LΦ, ϒ   5-4. Show for the heat transfer in a parallel plate channel that G G Φ j , Φ k = 0 for λ j ≠ λk by considering the eigenvalue problems G G G G LΦ j λ j j and d L k λk Φ k  for the case of constant wall temperature. 5-5. Derive from Eq. (5.69-5.70) the fully developed temperature distribution for large axial values for the case of laminar heat transfer in a pipe or a parallel plate channel with constant wall heat flux. Show that Eq. (5.75) is obtained in case of a circular pipe, whereas Eq. (5.76) is obtained for a parallel plate channel.

6 Nonlinear Partial Differential Equations

In the previous chapters, we discussed the solution of linear partial differential equations. Special focus was given to the solution of internal heat transfer problems in duct flows. However, in most technical applications, problems are often described by nonlinear partial differential equations. For a lot of these applications, the equations have to be solved by numerical methods. In contrast to the large amount of literature dealing with the solution of linear partial differential equations, much less literature exists on the solution of nonlinear partial differential equations. One of the major difficulties arising in the solution of nonlinear problems is that we are no longer able to use the powerful superposition method for constructing solutions as for linear problems. Sometimes, the equations under consideration may be linearised by using perturbation methods. An example on how to use this sort of method is shown in Chap. 4 for the solution of eigenvalue problems. In the present chapter, we do not discuss this solution approach. The interested reader is referred to the books of van Dyke (1964), Kevorkian and Cole (1981), Simmonds and Mann (1986), Aziz and Na (1984) and Schneider (1978) for many interesting applications of the perturbation method to fluid dynamics and heat transfer problems. In the following sections, we intend to provide a short overview on some selected solution methods for nonlinear partial differential equations occurring in heat transfer and fluid flow problems. The solution approaches discussed here include, for example, the method of separation of variables, the Kirchhoff transformation, and special solutions of the energy equation. However, the focal point of this chapter is on similarity solutions for the heat conduction and boundary layer equations. Here an overview is given on different methods and on how to determine these solutions. Similarity solutions are not only of importance because they may lead to analytical solutions of the underlying nonlinear partial differential equations, but also, because they play today an important role as excellent benchmark cases for testing computer codes.

6.1 The Method of Separation of Variables This solution method can also be used for nonlinear partial differential equations. However, since we can not use the superposition principle to fulfill arbitrary boundary conditions, we have to find a solution which already satisfies the partial differential equation and the related boundary conditions. The method is explained for the flow and heat transfer on a rotating disk, which rotates with the constant

166

6 Nonlinear Partial Differential Equations

angular velocity ω in a quiescent fluid. The geometry under investigation is depicted in Fig. 6.1

Fig. 6.1: Physical model and coordinate system for the rotating disk

Under the assumptions of angular symmetry for a laminar flow with constant fluid properties and negligible viscous dissipation, the conservation equations in cylindrical coordinates can be written in the following form (see for example Schlichting (1982)) Navier-Stokes Equations 2 ∂v 1 ∂p μ ª 2 v v vº =− + « 2+ + 2» ∂x ρ ∂r ρ ¬ ∂ ∂r © r ¹ ∂xx ¼

v

∂v ∂r

v

∂w wv ∂w + +u = ∂r r ∂x

v

w2 r

∂u ∂r

u

u

μ ρ

2 ª 2w wº § w· + 2» « 2 + ∂ r ∂ r r ∂x x © ¹ ¬ ¼

2 ∂u 1 ∂p μ ª 2 u 1 u uº =− + « 2+ + 2» ∂x ρ ∂x ρ ¬ ∂ r ∂r ∂xx ¼

(6.1)

(6.2)

(6.3)

Continuity Equation

∂v v ∂u + + =0 ∂r r ∂x

(6.4)

6.1 The Method of Separation of Variables

167

Energy Equation

v

2 § 2T 1 T ∂T ∂T T· +u = a¨ 2 + + 2¸ ∂r ∂x r ∂r ∂xx ¹ ©∂

(6.5)

For the rotating disk, the velocity components u, v and w denote the flow in the x, r and ϕ direction. Furthermore, we assume that the radius R of the rotating disk is very large ( R → ∞ ), so that no boundary conditions have to be satisfied for a finite value of the radius. The boundary conditions for the above equations are given by

x

0: u

x→

:v

0, v 0

00, w

r ,T

0, 0w

0, 0 T

T∞

(6.6)

TW

No boundary condition is provided for ( ) , because we only know that this quantity attains a finite value for x → ∞ . As stated above, we consider an incompressible fluid with constant fluid properties. Therefore, the energy equation is decoupled from the momentum equations and the continuity equation. This means that we can first solve the momentum equations and subsequently the energy equation. Let us assume that the solution to Eqs. (6.1-6.4) can be represented by the products (6.7)

v

A(( ) ( )

w

B(( ) ( )

u

C(( )

( )

From the boundary conditions given by Eq. (6.6), one immediately obtains w

B (r )G (0)

rω

(6.8)

From this equation it follows that B( ) B1 r , where B1 is a constant. If we select this constant to be equal to ω, it follows w

r ω G(( )),

(0) 1

(6.9)

After having gained some knowledge of the tangential velocity component w, let us introduce the expressions given by Eq. (6.9) into the tangential component of the Navier-Stokes equations, Eq. (6.2). This results in vω G(( )

( )

′( )

′′( )

μ/ρ

(6.10)

Introducing also Eq. (6.7) into Eq.(6.10) leads to ω A(( ) ( ) ( )

( ) ( ) ( )

( ) ( )

′( )

′′( ) μ / ρ

(6.11)

168

6 Nonlinear Partial Differential Equations

Comparing the left side of this equation to the right side, it is clear that Eq. (6.11) can only have a solution if the function A( ) A1 r and C(( ) C1 . Selecting the arbitrary constants A1 and C1 leads to A(r )

rω

(6.12)

μ/ρ

C(( )

This selection of the constants guarantees that the functions A, B and C have the same dimensions. Summarizing, the velocity field is given by v

rω F ( )

w

rω G(( )

u

ωμ / ρ

(6.13) ( )

Introducing the above expressions into the momentum equation in the x-direction results in

(

)

1 ∂p ρ ∂x

( ) ′( )

μ/ρ

′′( )

μ/ρ

(6.14)

From Eq. (6.14) it follows that ∂p = p′( ) ∂x

( )

(6.15)

( )

Furthermore, one sees from Eq. (6.14) that we can simplify this equation by introducing the following dimensionless coordinate

η=x

ωρ μ

(6.16)

Inserting Eqs. (6.15-6.16) into Eq. (6.14) results in H ( ) H ′( )

p ′ H ′′( ), )

p

p /( μ )

(6.17)

where the prime in Eq. (6.17) indicates the differentiation with respect to η. Introducing Eq. (6.13) and Eqs. (6.15-6.16) into Eq. (6.2) gives

ω 2 F 2 ω 2G 2

ωμ / ρ ω F ′ μ / ρω F ′′ = −

1 ′( ) ρ r

(6.18)

Because on the left hand side of this equation all quantities only depend on the axial coordinate, the right hand side of this equation must be a constant. This leads to the following expression for the function I(r)

6.1 The Method of Separation of Variables

I( )

r2 2

C1

169

(6.19)

C2

If we examine the pressure distribution for η → ∞ , the pressure needs to attain a uniform value. This means, that the constant C1 in Eq. (6.19) must be zero. The constant C2 can be included into ( ) given by Eq. (6.15). Thus, all expressions for the velocity components and the pressure are known. Introducing Eq. (6.13) and Eqs. (6.15-6.16) into the Eqs. (6.1-6.4) and the associated boundary conditions, Eq. (6.6), results in the following set of ordinary differential equations, which has to be solved for obtaining the velocity distribution over the rotating disk F 2 − G2

HF ′

F ′′

(6.20)

2FG HG ′ G ′′

(6.21)

HH ′ H

(6.22)

p ′ H ′′ ′

2

(6.23)

0

with the boundary conditions

η = 0: η →∞:

00, 0, 0

11, 0

0

(6.24)

The above set of equations has been solved first by von Karman (1921) by using an approximation method. Later Cochran (1934) presented a more accurate solution for the equations.

Fig. 6.2: Velocity distribution on a rotating disk (graph compiled from data reported by Cochran (1934))

170

6 Nonlinear Partial Differential Equations

Figure 6.2 shows the distribution of the functions F, F G and H as taken from data reported by Cochran (1934). From Fig. 6.2, it can be seen that the radial and circumferential velocity components decrease rapidly with increasing values of η, while the axial velocity component attains a finite value for large axial distances away from the rotating semi-infinite disk. Let us now focus on the heat transfer problem for the rotating semi-infinite disk. Introducing the known velocity distribution, Eq. (6.20-6.24), and the dimensionless coordinate, Eq. (6.16), into the energy equation results in Θ′′ = P Pr H Θ′

(6.25)

where the dimensionless temperature Θ=

(6.26)

T T∞ TW T∞

is used. Eq. (6.25) has to be solved together with the following boundary conditions

η = 0: Θ =1 η →∞:Θ = 0

(6.27)

The heat transfer problem on a rotating disk has been investigated by Millsaps and Pohlhausen (1952) for Prandtl numbers in the range of 0.5 to 10 and by Sparrow and Gregg (1959) for a large range of different values of the Prandtl number (0.01, 0.1, 1, 10 and 100). Eq. (6.25) has the general solution η § ξ Θ = C1 ³ exp ¨ Pr ¨ 0 © 0

· (ξ ))ddξ dξ ¸ ¹

(6.28) C2

The constants C1 and C2 can be obtained by satisfying the boundary conditions according to Eq. (6.27). This results in η

Θ =1

§ ξ exp ³0 ¨¨ Pr 0 ©

∞ · § ξ (ξ ) ξ ¸ ξ / expp ¨ Pr ¸ ¨ 0 ¹ © 0

· (ξ ))dξ dξ ¸ ¹

(6.29)

Sparrow and Gregg (1959) investigated the asymptotic behavior of Eq. (6.29) for small and large values of the Prandtl number. This analysis is shown here in some detail, because it elucidates nicely the behavior of the heat transfer for extreme values of the Prandtl number. Let us focus first on the case of very small molecular Prandtl numbers. Because the Prandtl number can be interpreted as the ratio of the thickness of the hydrodynamic to the thermal boundary layer (Kays and Crawford (1993)), a very small value of the Prandtl number indicates that the thermal boundary layer is much thicker than the hydrodynamic boundary layer. This means that for very small Prandtl numbers a constant value of the velocity distribution can be used as a first guess in order to solve the convective heat transfer problem. Replacing

6.1 The Method of Separation of Variables

therefore tains

171

( ) in Eq. (6.25), this equation can be integrated easily. One obPr

0:

exp(Pr

( ) ), )

( )

0.88447

(6.30)

The Nusselt number, defined by, h μ /( ρω ) Nu = = k

§ ∂T · −k ¨ μ /(( ¸ © ∂xx ¹ x = 0 k

)

(6.31) §d · = −¨ ¸ © dη ¹η = 0

where, in Eq. (6.31), the quantity μ /( ρ ) is used for the length scale in the expression of the Nusselt number. Inserting Eq. (6.30) into Eq. (6.31) results in Pr → 0 : Nu

0.88447 Pr

(6.32)

If the Prandtl number attains very large values, the thermal boundary layer is much thinner than the hydrodynamic boundary layer. This means that, in this case, the velocity distribution can be approximated by a Taylor series expansion of the velocity field around η = 0 . This leads to H (0)

H

1 2

′(0))

′′(0)

2

(6.33)

...

′(0) 0 , only the last term in Eq. (6.33) is retained. Inserting this Since (0) expression into Eq. (6.29) results after integration in η

Pr

:

1

§1

P ³ exp ¨© 6 Pr

3

((0))

0

· ¹

∞

/ exp e p 0

§1 Pr ©6

· (0))ξ 3 ¸ d ξ ¹

(6.34)

For the Nusselt number, one obtains from Eq. (6.34) and Eq. (6.31) (Sparrow and Gregg (1959))

Pr

:Nu

(

Pr

(0) / 6 )

1/ 3

/ (4 / 3)

0.62048 Pr1/ 3

(6.35)

The distribution of the Nusselt number, as a function of the Prandtl number, is shown in Fig. 6.3. The graph is based on calculated values of the Nusselt number from Sparrow and Gregg (1959). In this figure, the asymptotic distributions for small and large values of the Prandtl number are also plotted for comparison. It can be seen that there is a relatively good agreement between the asymptotic values and the numerically calculated Nusselt numbers for extreme values of the Prandtl number. Before concluding the present example, it is noteworthy stressing once again, where the difference in applying the method of separation of variables to linear and nonlinear partial differential equation lies. The striking difference comes from the fact that the superposition approach can no longer be successfully used. This

172

6 Nonlinear Partial Differential Equations

means that in contrast to Chaps. 2-5, boundary conditions can no longer be satisfied by adding up an infinite set of eigenfunctions.

Fig. 6.3: Nusselt numbers for a heated rotating disk for different Prandtl numbers (based on data from Sparrow and Gregg (1959))

6.2 Transformations Resulting in Linear Partial Differential Equations For some nonlinear partial differential equations, it is possible to reduce them to linear partial differential equations by using special transformations. Some of these transformations are discussed in Ames (1965a), Schneider (1978), Schlichting (1982), Loitsianki (1967), Carslaw and Jaeger (1992) and in Özisik (1968). For elucidating this type of solution approach, we discuss only one case of such a transformation. Consideration is given to the steady-state heat conduction within a two-dimensional region. The temperature field is described by the following heat conduction equation ∂ § ∂T · ∂ § T· ¨ ( ) ¸ ¨ k(( ) ¸ ∂xx © x ∂yy © ∂yy ¹

0

(6.36)

The heat conductivity in Eq. (6.36) is assumed to depend on temperature, but not on the coordinates x, y. This is a good assumption, which is sufficiently accurate for a large number of engineering problems. Since the heat conductivity is a function of temperature, the above equation is nonlinear. However, if we define a new dependent variable in the form,

6.2 Transformations Resulting in Linear Partial Differential Equations

T

Ω=³ 0

k((  )  dT k0

173

(6.37)

where k0 is a reference value of the thermal conductivity, Eq. (6.36) can be transformed into the linear Laplace equation ∂2Ω ∂ 2Ω + =0 ∂x 2 ∂y 2

(6.38)

The transformation given by Eq. (6.37) is known in literature as the Kirchhoff transformation. The only problem arising with this transformation is that also the boundary conditions have to be in a suitable form, so that after introducing Eq. (6.37) also the boundary conditions are linear. If we assume constant temperature boundary conditions, the transformation given by Eq. (6.37) works perfectly and the resulting equations are linear. The Kirchhoff transformation can also be applied to a much broader class of problems than the one shown above (see for example Carslaw and Jaeger (1992), Özisik (1968)). Let us consider one special example characterized by the following boundary conditions for Eq. (6.36) T(

T (x

y) T

)

T ( l y ) T0

(6.39)

T ( x l ) T1

T

where T0 and T1 are constant temperatures. Let us further assume that the heat conductivity within the material is given by the following equation k(( )

0

(1

(

0

))

(6.40)

where k0 and C are known constants. Introducing the dimensionless quantities Θ=

T T0 x y , x = , y = T1 T0 l l

(6.41)

into Eq. (6.36) and Eq. (6.39) results in

· § ( ) · ∂ § k(( ) ¨ ¸+ ¨ ¸=0 ∂xx © k0 x ¹ y © k0 y¹

(6.42)

with the boundary conditions Θ( Θ(

)= )=

Θ(

Θ(

If we now apply the Kirchhoff transformation

)=0 ) =1

(6.43)

174

6 Nonlinear Partial Differential Equations

Θ

Ω=³ 0

k((  )  ) dΘ  = Θ + C(( 1 d = ³ (1 dT ( + C (T1 − T0 )Θ k0 2 0 Θ

0

)

Θ2

(6.44)

to the Eqs. (6.42)-(6.43), we obtain ∂2Ω ∂ 2Ω + =0 ∂x 2 ∂y 2

(6.45)

and the corresponding boundary conditions

Ω(

)=

Ω(

)=0

Ω(

)=

Ω(

) = 1+

(6.46) C((

1

0

)

2

The above given problem for Ω can easily be solved by the method of separation of variables explained in Chap. 2. After obtaining the solution for the function Ω, the temperature distribution for the nonlinear heat conduction problem can be calculated from Eq. (6.44) and is given by Θ=

1 C((

1

0

( )

)

(6.47)

6.3 Functional Relations Between Dependent Variables Sometimes it is possible to establish functional relations between different dependent variables. This can be very useful. In case of the flow and heat transfer in a boundary layer it may be used to relate the heat transfer coefficient to the friction factor. 6.3.1 Incompressible Flow over a Heated Flat Plate

In order to elucidate this method, let us investigate the simple problem of a flow over a flat plate. It is assumed that the flow is incompressible and that the fluid properties are constant. The plate surface is kept at a uniform temperature TW , whereas the fluid far away from the wall moves with the constant velocity u∞ and has the constant temperature T∞ . The problem under consideration is depicted in Fig. 6.4

6.3 Functional Relations Between Dependent Variables

175

Fig. 6.4: Flow and heat transfer over a flat plate

The problem can be described by the boundary layer equations (Kays and Crawford (1993)) for a laminar flow with constant fluid properties u

u

∂u ∂u μ ∂ 2 u +v = ∂x ∂y ρ ∂yy 2

(6.48)

∂u ∂v + =0 ∂x ∂y

(6.49)

∂T ∂T +v ∂x ∂y

a

∂ 2T ∂yy 2

(6.50)

with the boundary conditions y

0: u

y

:u

00, v

00, T

TW

T

T∞

u ,

(6.51)

By comparing Eq. (6.48) to Eq. (6.50), it can be seen that both equations have a very similar structure. Introducing the following dimensionless quantities u =

T TW u v x y ,v = , x = , y = ,Θ = u∞ u∞ δ δ T∞ TW

(6.52)

into Eqs. (6.48-6.51), where δ denotes a length scale, one obtains u

∂∂uu ∂u μ ∂ 2 u + v = ∂x ∂y ρ u δ ∂yy 2

(6.53)

∂u ∂v + =0 ∂x ∂y

(6.54)

176

6 Nonlinear Partial Differential Equations

u

∂Θ ∂Θ a ∂2Θ + v = ∂x ∂y u∞δ ∂yy 2

(6.55)

with the boundary conditions y 0 : u 00,, v y → : u 1,

0, Θ

0

(6.56)

Θ 1

If we finally assume that the Prandtl number of the fluid is 1, which means that a = μ / ρ (which is a good assumption for all gases), one can see that Eq. (6.53) is identical to Eq. (6.55). In addition, the boundary conditions for u and Θ are the same. This means that both equations must have the same solution. Thus (6.57)

T TW u = T∞ TW u∞

Θ = u ,

Because the Prandtl number is equal to 1, the thickness of the hydrodynamic and the thermal boundary layer is the same in size. Equation (6.57) states the simple fact that the velocity profile in the boundary layer has in such a case the same shape as the temperature profile. Equation (6.57) is a very useful equation, because it can be used to establish a relation between the friction factor and the heat transfer coefficient at the surface. In order to show this, we will first predict the heat flux at the surface qW

k( § T ·§ u · k¨ ¸¨ ¸ = − u ∂y W u

§ T· k¨ ¸ © ∂yy ¹W

)§ ¨

u· ¸ ∂yy ¹W

(6.58)

Introducing the friction factor c f defined by cf =

2τ W § u· 2 = μ¨ ¸ 2 2 ρu © y ¹W ρ u∞

(6.59)

results in qW

k(

μ

) cf 2

ρ u∞

(6.60)

If we finally introduce the definition of the Nusselt number, based on the plate length L Nu L =

qW L hL = k k(

into the above equation, one obtains

(6.61)

)

6.3 Functional Relations Between Dependent Variables

177

(6.62)

cf Nu L = St = Re L Pr 2

Equation (6.62) states the interesting fact that the knowledge of the friction factor is sufficient for predicting the heat transfer for the above case. This relation is known as the Reynolds analogy in convective heat transfer. It is traditionally derived for a turbulent flow with Pr = 1 and Prt = 1 (see for example Kays and Crawford (1993)). The equation can be used in order to obtain an approximate value of the heat transfer coefficient by simply using the known friction factor. 6.3.2 Compressible Flow over a Flat, Heated Plate

Having established a direct relation between the friction factor and the heat transfer coefficient, one could ask whether similar expressions between the temperature field and the velocity field exist also for compressible flows. This is indeed the case and the following analysis should serve as another example for establishing functional dependences between dependent variables. Let us consider the case of a compressible flow over a flat plate. The equations describing the laminar flow and heat transfer in this situation are the compressible boundary layer equations (Kays and Crawford (1993), Schlichting (1982), Jischa (1982)). For the case of no axial pressure gradient (dp/dx = 0) in the flow, these equations are given by ∂ ( ∂x

)+

∂ ( ∂y

§

u u· +v ¸ ∂y ¹ © ∂x

ρ ¨u §

T ∂ © x

ρ cp ¨ u

v

T· ¸ ∂y ¹

)=0

(6.63)

§ u· ¨μ ¸ ∂y © ∂y ¹

(6.64)

§ T· § u· ¨k ¸+μ¨ ¸ ∂y © ∂y ¹ © ∂y ¹

2

(6.65)

In addition to the conservation equations, we have to prescribe for a compressible flow also a relation between p, ρ and T T. This is accomplished by the thermal state equation. If we assume that the fluid can be treated as an ideal gas, one has p = ρR RT

(6.66)

The above equations have to be solved with the boundary conditions y

0: u

y

:u

00, v u ,

00, T

TW

T

T∞

(6.67)

178

6 Nonlinear Partial Differential Equations

For a compressible flow, the Eqs. (6.63-6.66) are strongly coupled by the varying density. As in the case of the incompressible flow, we are looking for a functional relationship of the form T

(6.68)

f (u )

Inserting Eq. (6.68) into the energyy equation, Eq. (6.65), results in dT § u u ddu © x

ρcp

v· y¹

v

§

y©

k

§ u· u dT · ¸+μ¨ ¸ y ddu ¹ © y¹

2

(6.69)

This equation can be further simplified using Eq. (6.64) and one obtains cp

dT ∂ § u · ¨μ ¸ du ∂yy © y¹

§ u dT · § u· ¨k ¸+μ¨ ¸ y © y ddu ¹ © y¹

2

(6.70)

Performing all differentiations on the right hand side of Eq. (6.70) leads to cp

dT ∂ § u· ¨μ ¸ du ∂yy © y¹

§ u· dT d 2T ¨k ¸+ k 2 du d y© y¹ ddu

2

§ u· § u· ¨ ¸ +μ¨ ¸ ∂ y © ¹ © ∂y ¹

2

(6.71)

Rearranging gives finally dT § u μ ¨ cp du © y© y¹

y©

k

u · y ¹¹

2

· § u · § d 2T k 2 +μ¸ y du d © ¹ © ¹

(6.72)

Let us now assume that c p is constant. Then we introduce the Prandtl number ( Pr

μ

p

/ k ) into Eq. (6.72). This leads to dT § ∂ § u kP Pr ¨ du © y © y¹

y©

k

u · ¸ y ¹¹

2

· § u · § d 2T ¨ ¸ ¨k 2 + μ ¸ © y ¹ © ddu ¹

(6.73)

For Pr = 1, Eq. (6.73) simplifies drastically and one obtains k

(6.74)

d 2T +μ =0 du 2

Equation (6.74) is an ordinary differential equation, which can be integrated easily. Adapting the solution to the boundary conditions given by Eq. (6.67) leads to T TW u∞2 u u = + T∞ TW u∞ 2c p (T∞ TW ) u∞

§ u · 1 ¨1− ¸ © u∞ ¹

(6.75)

Equation (6.75) shows nicely that the linear dependence between temperature and velocity, see Eq. (6.57), is now replaced by a nonlinear relationship. For small velocities, the quadratic term on the right hand side of Eq. (6.75) can be neglected

6.4 Similarity Solutions

179

and one obtains again Eq. (6.57). However, if we rewrite Eq. (6.75) using the stagnation temperatures instead of static temperatures, which means T0

T

1 u2 T0 2 cp

T

1 u2 2 cp

T0W

TW

(6.76)

one obtains again T0 T0W u = T0 T0W u∞

(6.77)

This is an extension of the similarity between the velocity distribution and the temperature distribution given by Eq. (6.57). For a more detailed discussion of this case, the reader is referred to Schlichting (1982) and Loitsianki (1967).

6.4 Similarity Solutions In the following section, we are interested in “similar solutions” of nonlinear partial differential equations. Mathematically speaking, similarity solutions of partial differential equations appear when the number of independent variables can be reduced by a transformation of these variables. If we focus on the important case of a partial differential equation with two independent variables, this means that an ordinary differential equation is obtained after the transformation. Similarity solutions play an important role in boundary layer theory and also in heat conduction. For example, in boundary layer theory, similarity solutions can be obtained for some cases if the physical problem under consideration is lacking of a characteristic length or a characteristic time scale (e.g. the boundary layer on a semi-infinite plate with zero pressure gradient). In such cases, the nonlinear partial differential equations for the boundary layer are reduced to nonlinear ordinary differential equations. There are several books dealing with similarity solutions and the different methods to obtain them. The reader is referred for example to Hansen (1964), Ames (1965a), Schneider (1978), Dewey and Gross (1967) and Özisik (1968) for detailed discussions on similarity solutions. Basically, there are three main classes of methods for obtaining similarity solutions (Hansen (1964)). These three methods are explained in the following section for a simple heat conduction problem. Subsequently, we analyse similarity solutions for different boundary layer problems. 6.4.1 Similarity Solutions for a Transient Heat Conduction Problem

Consider the one-dimensional transient heat conduction in a semi-infinite solid. The body has a uniform initial temperature T0. For 0 , the wall temperature is

180

6 Nonlinear Partial Differential Equations

set to the constant temperature TW. The properties of the solid are assumed to be constant. The problem is described by the heat conduction equation for the solid

ρc

∂T ∂t

∂ § ∂T · ¨k ¸ ∂x © ∂xx ¹

(6.78)

with the boundary conditions t

0 : T T0

x

0 : T TW ,

(6.79) x→

:T

T0

The problem under consideration is depicted in Fig. 6.5. Of course, Eq. (6.78) represents a linear partial differential equation and can be solved also with other methods. However, the simplicity of this equation makes it a good candidate to explain the different methods for obtaining similarity solutions, before considering more complicated, nonlinear problems.

Fig. 6.5: Transient heat conduction in a semi-infinite solid

Methods Based on Dimensional Analysis

The dimensional analysis is a powerful method, which can be used for all physical applications. The method is based on the fact, that all physical quantities have units, which can be split into “basic units”. These basic units could be, for example, the international SI-system with the units: Length L (m), mass M (kg), time τ (s), temperature ϑ (K), species amount N (mol), current strength I (A) and light strength S (cd). The density for example has the unit mass M divided by length cubed L3, which can be denoted by [ML-3]. In general, one can say that for any problem there is always the task to express a physical quantity p1 as a function of other physical quantities p2 p3 ,..., pn , where n is the number of physical variables needed to describe the problem under consideration (Görtler (1975), Spurk (1992), Simon (2004)). This means that a relation f ( p1 , p2 ,..., pn )

0

(6.80)

6.4 Similarity Solutions

181

can be established. Each of the physical variables p j has units, which can be constructed from the basic units stated above. This means that the dimension of the variable p j , denoted by ª¬ p j º¼ , is given by m

Π(

ª¬ p j º¼

i =1

)

i

(6.81)

aij

where X i denotes one of the basic units powered by the exponent aij . For the above example of the density, one has

[ ] = M 1L 3 ,

11,

1

3

2

(6.82)

Since every physical process must be independent of the arbitrary units used, the relation given by Eq. (6.80) can be reduced to a relationship between dimensionless quantities Π j , so that.

f(

1

,

2

,...,

d

)

0

(6.83)

where d is the number of dimensionless products. These dimensionless quantities are products of the physical quantities, which means

∏ = ∏(

)

n

(6.84)

bj

j =1

with the dimension n

ª¬ p j º¼ [ ] =1= ∏ j =1

bj

(6.85)

Inserting Eq. (6.81) into Eq. (6.85) leads to n

m

a b ∏( ) [ ] =1= ∏ j 1i 1

ij j

(6.86)

In Eq. (6.86) the coefficients aij are known and the exponents b j have to be determined, so that the equation is satisfied. This is always possible, if the system of equations n

¦a b ij

j

=0

(6.87)

j =1

has nontrivial solutions for the n unknown coefficients bj. The coefficient matrix aij is also known as the dimension matrix. The equivalence between Eq. (6.80) and Eq. (6.83) is known in literature as the “Buckingham theorem” or as the “Pi theorem” (Görtler (1978), Spurk (1992), Simon (2004)). This theorem gives also a relation between the number of physical

182

6 Nonlinear Partial Differential Equations

variables involved, n, and the number of independent dimensionless groups, d, according to d

(6.88)

n r

where r is the rank of the dimension matrix. Let us now focus again on the problem of finding similarity solutions for the Eqs. (6.78-6.79). The physical quantities involved in the problem are: ρ , c, k , T0 , TW , T , t , x . This means that we can construct the following dimension matrix Table 6.1: Dimension matrix for the transient heat conduction problem

L M

τ

ϑ

ρ

c

k

T T0

T0 TW

t

x

-3 1 0 0

2 0 -2 -1

1 1 -3 -1

0 0 0 1

0 0 0 1

0 0 1 0

1 0 0 0

where the units for the heat conductivity k ª¬Wm W 1 K 1 º¼ and for the specific heat c ª¬ J kg kg 1 K 1 º¼ have been expressed in basic units. Note that, in the dimension matrix, we have used only temperature differences. The rank of the above dimension matrix r = 4 , which implies that the number of dimensionless products is d n r = 7 4 3 . However, upon inspection of Eq. (6.78), one notices that the quantities ρ , c, k do not appear independently in this equation. They can be lumped together to build the heat diffusivity /( ρ ) . By doing so, the dimension matrix simplifies to Table 6.2: Simplified dimension matrix for the transient heat conduction problem

L M

τ

ϑ

a

T T0

T0 TW

t

x

2 0 -1 0

0 0 0 1

0 0 0 1

0 0 1 0

1 0 0 0

In this matrix, no entries appear for the mass M M, so that the rank of this matrix is only 3. This means that we can construct 2 dimensionless quantities. By doing so, one obtains

6.4 Similarity Solutions

183

Table 6.3: Dimension matrix with dimensional groups

x / at L M

τ

ϑ

(

0

0 0 0 0

) /(

0

)

T0 TW

t

x

0 0 0 1

0 0 1 0

1 0 0 0

0 0 0 0

This means that the solution of the current problem must have the functional form T T0 § x · = f¨ ¸ TW T0 © att ¹

(6.89)

Introducing the new coordinate

η=

x

(6.90)

at

into Eqs. (6.78-6.79), results in the ordinary differential equations 1 Θ′′ + η Θ′ = 0 2

(6.91)

where the prime denotes differentiation with respect to the coordinate η . The boundary conditions for this equation are

η = 0: Θ =1 η → ∞ :Θ = 0

(6.92)

Equation (6.92) shows nicely that the two boundary conditions for t = 0 and for x → ∞ given by Eq. (6.79) have been replaced by one boundary condition for η → ∞ . Eq. (6.91) together with the boundary conditions given by Eq. (6.92) can be integrated and one finally obtains Θ = 1−

η

§ η2 · exp p ³ ¨© − 4 ¸¹ dη π 0

1

(6.93)

Group-Theory Methods

This method for obtaining similarity solutions has been suggested by Birkhoff (1948) and has been investigated extensively by Morgan (1951), Müller and Matschat (1962), Aimes (1965a) and Aimes (1965b). The method implies that the search for similarity solutions for a system of partial differential equations is equivalent to the determination of the invariant solutions of these equations under a particular parameter group of transformation. A set of similarity variables, which

184

6 Nonlinear Partial Differential Equations

are invariants of the group, is obtained by solving a set of simultaneous algebraic equations. Suppose Γ is a set of N partial differential equations given by Γ :ψ j ( i,

j

)

0, 0

i 1, 2,3,..., M

(6.94)

j = 1, 2,3,..., N where

i

,

1, 2,3,..., M are the independent and y j , j 1, 2,3,..., N are the de-

pendent variables. Now we are considering a one-parameter group G1 with the nonzero real parameter A, whose elements are given by xi

Aγ i xi

yj

A j yj

(6.95)

κ

In Eq. (6.95) A is a nonzero real number, called the parameter of the transformation, and the exponents γ i , κ j have to be determined from the requirement that the system of partial differential equations, given by Eq. (6.94), is absolutely invariant under the transformation given by Eq. (6.95). This means that

ψ j( i,

j

)

j

( i,

j

(6.96)

)

After applying the transformation to the partial differential equations, one obtains a set of simultaneous algebraic equations for determining the unknown coefficients γ i and κ j . More detailed information on this method can be found in the literature mentioned above. Let us illustrate the method by obtaining similarity solutions for the Eqs. (6.786.79). We assume a linear transformation for the dependent and independent variables in Eq. (6.76) of the form x

Aγ 1 x, t

T

Aκ1 T

Aγ 2 t

(6.97)

As noted before γ 1 γ 2 κ 1 are constants and the quantity A is called the parameter of the transformation. Solving the above given equation for A gives 1/ γ 1

§x· A=¨ ¸ © x¹

1/ γ 2

§t · =¨ ¸ ©t¹

1/ κ1

§T · =¨ ¸ ©T ¹

(6.98)

Introducing the quantities given by Eq. (6.97) into Eq. (6.78) results in Aγ 2

κ1

ρc

∂T ∂t

A2 1

1

∂ § T· ¨k ¸ ∂x © ∂xx ¹

(6.99)

Comparing Eq. (6.99) to Eq. (6.78) shows clearly that the partial differential equation does not get altered by the transformation if

6.4 Similarity Solutions

κ 1 − γ 2 = κ 1 − 2γ 1

185

(6.100)

This equation leads immediately to the result that γ 2 2γ 1 , whereas nothing can be said about κ 1 . This is obvious, because the differential equation is linear and therefore every exponent for the temperature drops out. Introducing the relationship between γ 1 and γ 2 into Eq. (6.98), it can be seen that the coordinates should be related in the following way in order to guarantee similarity solutions 1/ γ 1

§x· ¨ ¸ ©x¹

1/ 2 γ 1

§t · =¨ ¸ ©t¹

=> η =

t

This new coordinate can be made dimensionless by using tains the similarity coordinate given by Eq. (6.90)

η=

(6.101)

x

/( ρ ) and one ob-

x

(6.90)

at

Introducing again the dimensionless temperature Θ = ( − ) /( 0 ) leads after performing the change of variables in Eq. (6.78), to the ordinary differential equation (6.91) with boundary conditions given by Eq. (6.92). Separation of Variables and the Method of the Free Variable

Both methods are strongly related to each other. We have already seen in section 6.1 how the method of separation of variables can be used in order to find a similarity solutions for partial differential equations. Here we want to investigate the method of the free variable (see also Schneider (1978)). For this method, we simply transform the coordinates in Eq. (6.78) according to

τ =t η = x g (t )

(6.102)

The new variable η contains the unknown function g ( ) . This variable is the free variable, which gave the method its name. In order to use the method for obtaining similarity solutions for Eqs. (6.78-6.79), we introduce the dimensionless temperature distribution Θ = ( − ) /( 0 ) into these equations. This results in ∂Θ ∂2Θ =a 2 ∂t ∂x

(6.103)

with the boundary conditions t = 0: Θ=0 x = 0 : Θ =1,

(6.104) → ∞:Θ = 0

186

6 Nonlinear Partial Differential Equations

Introducing the new coordinates defined in Eq. (6.102) into the partial differential equation (6.103) results in

η d g ′( ) g( ) dη

a g2( )

d 2Θ dη 2

(6.105)

where the prime denotes differentiation with respect to τ . In Eq. (6.105) we have assumed that Θ depends only on η. This assumption is suggested by the form of the boundary conditions given by Eq. (6.104). Eq. (6.105) reduces to an ordinary differential equation, if all dependence on the time drops out of this equation. This is satisfied, if g ′( ) = const. g3 ( )

(6.106)

From this equation we can calculate g(( ) g ( ) 1/ C1 aτ

(6.107)

The constant C1 in Eq. (6.107) can be set to zero, because it only shifts the time origin and we are only interested in one solution of Eq. (6.106). Introducing Eq. (6.107) with C1 = 0 into Eq. (6.105) leads to 1 Θ′′ + η Θ′ = 0 2

(6.91)

which is identical to Eq. (6.91). 6.4.2 Similarity Solutions of the Boundary Layer Equations for Laminar Free Convection Flow on a Vertical Flat Plate

After discussing the fundamentals of the different methods for a very simple example, the present section investigates similarity solutions of the boundary layer equations for laminar, incompressible, free convection flow on a vertical flat wall. The reader is referred to Jaluria (1980) and Ede (1967) for detailed discussions on this topic. Similarity solutions for this type of application have been obtained for example by Ostrach (1953) which are in very good agreement with measurements performed by Schmidt and Beckmann (1930). The geometry as well as the velocity distribution for the free convection on a vertical flat plate is shown in Fig. 6.6. The free convection flow is driven by the temperature difference between the wall and the free-stream. It is assumed that the Bousinesq approximation can be applied. This means that the density difference can be replaced by a simplified equation of state and that variable property effects can be neglected, except for the density in the momentum equation (see for example Jaluria (1980)).

6.4 Similarity Solutions

187

Fig. 6.6: Free convection flow on a vertical flat plate

In the above equation, g is the gravitational acceleration and β denotes the thermal volumetric expansion coefficient, defined by

β=

1 § ∂ρ · ρ ¨© ∂T ¸¹ p

(6.109)

The continuity equation (6.49) can automatically be satisfied by introducing a stream function, defined by u

∂Ψ ∂Ψ , v=− ∂y ∂x

(6.110)

This leads to the following set of partial differential equations ∂Ψ ∂ 2 Ψ ∂Ψ ∂ 2 Ψ − = gβ ( ∂y ∂x∂y ∂x ∂y 2

−

)Θ +

∂Ψ ∂Θ ∂Ψ ∂Θ ∂2Θ − =a 2 ∂y ∂x ∂x ∂y ∂yy with boundary conditions

μ ∂3Ψ ρ ∂y 3

(6.111)

(6.112)

188

6 Nonlinear Partial Differential Equations

y 0 : ∂Ψ ∂y y → : ∂Ψ ∂y

0,, ∂Ψ ∂x 0,

0, Θ 1 Θ 0

(6.113)

and the dimensionless temperature Θ=

(6.114)

T T∞ TW T∞

Group-Theory Methods

The group-theory method has been used, for example, by Ames (1965a) and by Cebeci and Bradshaw (1984) for obtaining similarity solutions for the boundary layer equations for forced convection flows. Here we want to investigate the free convection on a vertical plate by using the linear transformation x

Aγ 1 x,

Aκ1 Ψ

y

Aγ 2 y,

Aκ 2 Θ

(6.115)

for the dependent and independent variables. From Eq. (6.115) one obtains 1/ γ 1

§x· ¨x¸ ©

1/ γ 2

§y· =¨ ¸ © y¹

1/ κ1

§ · =¨ ¸ ©Ψ¹

1/ κ 2

§ · =¨ ¸ ©Θ¹

(6.116)

=A

Introducing the linear transformation, given by Eq. (6.115), into the Eqs. (6.1116.112) results in Aγ 1 +22

2

g β (TW

Aγ 1 +γ 2

κ1 κ 2

2κ1

∂Ψ ∂ 2 Ψ ∂y ∂x ∂y

T ))A A

∂Ψ ∂Θ ∂y ∂x ∂y

κ2

Aγ 1 +γ 2

∂Ψ ∂ 2 Ψ = ∂x ∂y 2

Aγ 1 +22γ 2

2κ 1

A3γ 2

μ ∂3Ψ ρ ∂y 3

κ1 κ 2

κ1

∂Ψ ∂Θ ∂x ∂y

aA2γ 2

κ2

(6.117)

∂2Θ ∂y 2

(6.118)

If one requires that the differential equations are completely invariant to the proposed linear transformations, the following coupled algebraic equations are obtained

γ 1 + 2 2 − 2κ 1 = −κ 2 3γ 2 κ 1 κ2 γ 1 + γ 2 − κ1 − κ 2 = 2γ 2 − κ 2

(6.119)

6.4 Similarity Solutions

Under the assumption that the exponent γ 1 ≠ 0 , a ratio γ Thus one obtains from Eq. (6.119) the following relations

γ 2 κ1 , γ1 γ1

γ

κ2 γ1

1 γ,

189

γ 2 / γ 1 can be defined. (6.120)

1 4γ

Introducing Eq. (6.120) into Eq. (6.116) results in

η=

y , xγ

1

1 4γ

( ), )

(6.121)

( )

with the yet unknown coefficient γ . However, if one introduces the expression for the temperature into the boundary conditions, Eq. (6.113), one obtains

η = 0: η →∞:

1 4γ

(6.122)

(0) 1

1 4γ

( )

0

From the boundary condition for η → ∞ it can be seen that ( ) 0 . From the second boundary condition for η = 0 it is obvious that the exponent γ must be 1/4. Inserting this value of γ into Eqs. (6.121) yields

η=

y 1/ 4

x

,

3/ 4

( ),

(6.123)

( )

Transforming the individual terms into Eqs. (6.111-6.113) results in

§ · § ∂Ψ · 1/ 2 ¨ ¸ = x F ′( )), ¨ ¸ x ¹y © ∂yy ¹ x

3 x 4

1/ 4

§ 2 · § 2 · 1 1/ 4 ′′ = x F ( ), ) ¨ 2 ¸ ¨ ¸= x © ∂yy ¹ x © ∂x∂y ¹ 2

1 4

F( )

1/ 2

1 4

F( )

(6.124)

′( )),

1/ 4

1/ 2

′( )),

§ 3 · ¨ 3 ¸ = F ′′′( ) © ∂yy ¹ x where a prime in Eq. (6.124) indicates a differentiation with respect to the similarity coordinate η . Inserting these expressions into Eqs. (6.111-6.113) results with

ν

μ / ρ in −

1 ( 2

)

2

3 (η ) ′′(η ) ν ′′′′ (η ) g β ( 4 3 F ( ) ′( ) 4

with the boundary conditions

′′( )

0

W

) (η )

0

(6.125)

(6.126)

190

6 Nonlinear Partial Differential Equations

η = 0 : (0) 00, ′(0) 0, 0 (0) 1 η → ∞ : ′( ) 0, ( ) 0

(6.127)

Introducing finally the following dimensionless quantities

η

y § gβ ( W 1/ 4 ¨ x © ν2

F

(

§ β( W ν2 ©

1/ 4

)· ¸ ¹

)

1/ 4

1/ 4

)· ¸ ¹

η¨

(6.128)

F

into Eqs. (6.125-6.127) gives −

1 2

(

)

3  ( )  ′′( ) 4

2

3 Pr  ( ) ′( ) 4  (0)

η = 0 :

 ′′′( )

′′( )

0, 0  ′(0)

η → ∞ :  ′( ) 0,

( )

0

0

0, 0  (0) 1 ( ) 0

(6.129)

(6.130)

(6.131)

The Method of the Free Variable

Let us now investigate the same problem, Eqs. (6.111-6.113), with the method of the free variable. Therefore, we introduce the new coordinates, defined by

ξ=x η = y g ( x)

(6.132)

In addition, we assume that the stream ffunction can be prescribed by the following expression Ψ = H (ξ ) ( )

(6.133)

Introducing the new coordinates Eq. (6.132) and the stream function, given by Eq. (6.133), into Eqs. (6.111-6.113) results in

( ) ( H ′g (ξ ) 2

g ′(ξ ) ) g (ξ )

ν g 3 (ξ ) HF ′′′ g β (

W

(6.134)

)Θ

§ H′ · a Θ′′ + F ¨ ¸ Θ′ = 0 © g (ξ ) ¹ with the boundary conditions

′′g 2 (ξ ))HH HH ′ =

(6.135)

6.4 Similarity Solutions

η = 0: = 0, ′ = 0, Θ = 1 η → ∞ : ′ = 0, Θ=0

191

(6.136)

In order to reduce Eqs. (6.134-6.135) to a set of ordinary differential equations, the dependence on the coordinate ξ has to drop out from these equations. This leads to the fact that the following ordinary differential equations have to be satisfied Hg ( H g H

Hg

)

C1

g 2 HH ′

(6.137)

C2

3

g H H′ g

C3 = C4

where C1 2 3 andd C4 are constants. The four differential equations, given by Eq. (6.137) have to be linearly dependent on each other so that the two functions H and g can be determined uniquely. If we divide, for example, the second relation in Eq. (6.137) by the third one, the fourth relation is obtained. If we eliminate H and H ′ in the second relation using the third and the fourth relations, one obtains C3C4

(6.138)

C2

Introducing the third and the fourth relations into the first one results in the following ordinary differential equation for the unknown function g g ′ § C1 C4 · =¨ − ¸ g 5 © C32 C3 ¹

(6.139)

This ordinary differential equation has the solution

(

g

)

−1/ 4

(6.140)

where B1 and B2 are constants. The constant B2 can be set to zero, because it only shifts the origin of the coordinate. For the function H one obtains from Eq. (6.137) H

B3ξ 3 / 4

(6.141)

Introducing Eqs. (6.140-6.141) into the Eqs. (6.132-6.133), one obtains for the similarity coordinate and for the stream function

η=

y 1/ 4

x

,

3/ 4

( )

(6.142)

192

6 Nonlinear Partial Differential Equations

This are the same expressions which have been derived by using the group-theory method (see Eq. (6.123)). 6.4.3 Similarity Solutions of the Compressible Boundary Layer Equations

As a final example, we want to consider the compressible boundary layer equations for laminar, forced convection flow over a flat plate. Similarity solutions for compressible flows have been investigated for example by Li and Nagamatsu (1953, 1955). A good overview on this subject is provided for example by Schlichting (1982). For a compressible, laminar flow, the boundary layer equations take the form (Kays and Crawford (1993)) ∂ ( ∂x

ρu

∂u ∂xx

ρv

T ∂x © x

v

T· ¸ y¹

§

ρ cp ¨ u

)+

∂ ( ∂y

∂u y

u

dp p ddx dpp dx d

(6.63)

)=0 ∂ § u· μ ¸ y© y¹

§ T· § u· ¨k ¸ μ¨ ¸ y© y ¹ © y¹

(6.143)

2

(6.144)

As it can be seen from the system of Eqs. (6.63, 6.143-6.144), they are strongly coupled by the density. In addition, the fluid properties may no longer be considered constant, because of the high velocities under consideration. It can be assumed that the dynamic viscosity can be approximated by

μ(

μ(

)

)

§ T =¨ ¨T © rref

ω

· ¸¸ , ¹

(6.145) 0.5

1

where Tref is a reference temperature. The density is related to the pressure and the temperature by a thermal state equation and we assume that the fluid under consideration can be considered as an ideal gas.

ρ=

p RT

(6.146)

In addition, we assume that cp is constant. For the compressible flow, it is preferable to introduce the total enthalpy into the boundary layer equations. The total enthalpy is defined for a boundary layer flow ( v